Foundation

What Are Multiple Angle Formulas — and Why Are They Their Own Topic?

Core Idea

Multiple angle formulas express the trigonometric ratios of multiples of an angle — sin 2A, cos 3A, tan 2A — entirely in terms of the ratios of the original angle A. Sub-multiple angle formulas do the reverse: they express ratios of A in terms of ratios of A/2. Both families come from the same source — the compound angle (addition) formulas — with a single substitution. That’s the key insight the whole topic turns on.

Students sometimes treat these formulas as a list to memorise. That’s the wrong approach. If you understand where each formula comes from — which addition formula it derives from, which Pythagorean identity it uses — you can reconstruct any of them on the spot. You don’t need to memorise 12 separate facts. You need to understand about 3.

The topic is significant because these identities appear in almost every branch of further mathematics: integration by substitution, differential equations, Fourier analysis, physics wave equations, and signal processing all rely on double and half angle forms. In exams — JEE, A-Level, Class 12 board papers, and undergraduate calculus — multiple angle questions test whether you can manipulate expressions fluently, not just recall formulas.

3 Forms of cos 2A to know
2 Source formulas everything derives from
1 Substitution (B = A) that creates all double angle formulas
∞ Applications across calculus, physics, engineering
2️⃣

Double Angle

sin 2A, cos 2A (3 forms), tan 2A — derived from compound angle with B = A

3️⃣

Triple Angle

sin 3A, cos 3A, tan 3A — derived by applying compound formulas again

½

Sub-Multiple (Half Angle)

sin A/2, cos A/2, tan A/2 — same identities with A replaced by A/2

🔁

Product-to-Sum

Expressing products like sin A cos A as sums — follows directly from double angle forms

Derivation First

Where They Come From — The Two Formulas That Produce Everything

Before you look at a single double angle formula, you need to have these two compound angle identities solid. Every multiple angle result in this topic is a consequence of one of them.

The Two Source Formulas Compound Addition Formula — Sin:
   sin(A + B) = sin A cos B + cos A sin B

Compound Addition Formula — Cos:
   cos(A + B) = cos A cos B − sin A sin B

Compound Addition Formula — Tan:
   tan(A + B) = (tan A + tan B) / (1 − tan A tan B)

Now set B = A in each of these. That one substitution generates all double angle formulas.

That’s the trick the whole topic is built on. Set B = A. Every double angle formula is just the compound formula evaluated at a specific point. Students who grasp this — rather than treating double angle formulas as separate facts — understand the topic at a level that lets them adapt when questions are presented in unfamiliar forms.

For triple angle formulas, the approach extends naturally. Write sin 3A = sin(2A + A), then apply the addition formula again, substituting your double angle results for sin 2A and cos 2A. Each step is just the same tool applied one more time.

📌

The Pythagorean Identity Connects Everything

The reason cos 2A has three equivalent forms is the Pythagorean identity: sin²A + cos²A = 1. Once you have cos 2A = cos²A − sin²A from the compound formula, you can substitute sin²A = 1 − cos²A to get 2cos²A − 1, or cos²A = 1 − sin²A to get 1 − 2sin²A. These are not three separate formulas — they are one formula in three different arrangements, each useful in a different type of problem.

Core Family 1

Double Angle Formulas — The Full Set and How to Derive Each One

sin

sin 2A

One form. Clean and direct.

sin 2A = 2 sin A cos A
Derivation: sin(A+A) = sinA cosA + cosA sinA
Also: sin 2A = 2 tan A / (1 + tan²A)
cos

cos 2A — 3 Forms

One identity, three arrangements.

cos 2A = cos²A − sin²A
cos 2A = 2cos²A − 1
cos 2A = 1 − 2sin²A
Also: (1 − tan²A)/(1 + tan²A)
tan

tan 2A

From tan addition formula.

tan 2A = 2 tan A / (1 − tan²A)
Derivation: tan(A+A) with compound formula
Undefined when tan²A = 1 (i.e. A = 45°, 135°…)

Which form of cos 2A should you use?

This is the most common point of confusion. The answer is: it depends on what else is in the expression. Pick the form that eliminates a ratio you don’t want.

If the expression contains… Use this form of cos 2A Why
Only cos terms 2cos²A − 1 Eliminates sin²A, leaves everything in cos
Only sin terms 1 − 2sin²A Eliminates cos²A, leaves everything in sin
Both sin and cos cos²A − sin²A Most direct; symmetric form works for both
tan terms throughout (1 − tan²A)/(1 + tan²A) Converts everything to tan for a uniform expression
💡

Rearranging cos 2A Is a Standalone Skill

Two of the most useful algebraic tools in this topic come from rearranging the cos 2A forms. From cos 2A = 2cos²A − 1, you get cos²A = (1 + cos 2A)/2. From cos 2A = 1 − 2sin²A, you get sin²A = (1 − cos 2A)/2. These two rearrangements are exactly what you need to integrate sin²x and cos²x — and they appear constantly in calculus and physics. Know them cold.

Core Family 2

Triple Angle Formulas — Building from Double Angle Results

Triple angle formulas are not memorised from scratch. You derive them by writing 3A = 2A + A and applying the addition formula. The double angle results you already know feed directly into the derivation. Here’s how the reasoning flows for sin 3A:

Deriving sin 3A — Step by Step Logic Step 1: Write 3A as a sum
   sin 3A = sin(2A + A)

Step 2: Apply compound addition formula
   = sin 2A · cos A + cos 2A · sin A

Step 3: Substitute double angle formulas
   = (2 sin A cos A) · cos A + (1 − 2sin²A) · sin A

Step 4: Expand and simplify
   = 2 sin A cos²A + sin A − 2sin³A

Step 5: Replace cos²A using Pythagorean identity
   = 2 sin A (1 − sin²A) + sin A − 2sin³A
   = 2 sin A − 2sin³A + sin A − 2sin³A

∴ sin 3A = 3 sin A − 4sin³A
Deriving cos 3A — Same Approach Write cos 3A = cos(2A + A) and apply compound formula:
   cos 3A = cos 2A · cos A − sin 2A · sin A
   = (2cos²A − 1) · cos A − (2 sin A cos A) · sin A
   = 2cos³A − cos A − 2sin²A cos A
   = 2cos³A − cos A − 2(1 − cos²A) cos A
   = 2cos³A − cos A − 2cos A + 2cos³A
∴ cos 3A = 4cos³A − 3 cos A
Key Result

sin 3A = 3 sin A − 4 sin³A

Notice the pattern: a linear term minus a cubic term. The coefficient on the cubic is 4 (same for cos 3A). Remembering this pattern — rather than the full formula — is often enough to reconstruct it.

Key Result

cos 3A = 4 cos³A − 3 cos A

Same structure: cubic term minus linear term (coefficients 4 and 3 respectively). The derivation logic is identical to sin 3A — write 3A = 2A + A and apply compound then double angle formulas.

tan 3A — Derived from Addition Formula tan 3A = tan(2A + A)
   = (tan 2A + tan A) / (1 − tan 2A · tan A)
   Substitute tan 2A = 2 tan A / (1 − tan²A) and simplify:

tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)
Again: cubic numerator, quadratic denominator. Same family of structure as sin 3A and cos 3A.
🔢

The Reverse Direction — Converting Cubic Terms Back to Multiple Angles

Triple angle formulas are also used in reverse. You’ll often see exam questions that present sin³A or cos³A and ask you to simplify. Rearrange the triple angle results:

  • sin³A = (3 sin A − sin 3A) / 4  (rearranging sin 3A formula)
  • cos³A = (3 cos A + cos 3A) / 4  (rearranging cos 3A formula)

These are useful for integrating powers of trig functions — ∫sin³A dA becomes straightforward once you replace sin³A with this form.

Core Family 3

Sub-Multiple Angle Formulas — How Half-Angle Forms Work and When to Use Them

Sub-multiple angle formulas look like a separate topic. They’re not. They are the double angle formulas with A replaced by A/2. That’s the whole move. Once you see that, the entire set of half-angle formulas falls out immediately.

Getting Sub-Multiple Formulas from Double Angle Formulas In every double angle formula, replace A with A/2:

sin 2(A/2) → sin A = 2 sin(A/2) cos(A/2)

cos 2(A/2) → cos A = cos²(A/2) − sin²(A/2)
                          = 2cos²(A/2) − 1
                          = 1 − 2sin²(A/2)

tan 2(A/2) → tan A = 2 tan(A/2) / (1 − tan²(A/2))

These are the sub-multiple (half angle) forms. Not new identities — same identities, new variable.

Expressing sin(A/2), cos(A/2), tan(A/2) explicitly

Rearranging the cos A forms (which involve A/2) gives you explicit expressions for the half-angle ratios themselves:

Half-Angle Explicit Forms From cos A = 1 − 2sin²(A/2):
   sin(A/2) = ±√[(1 − cos A)/2]

From cos A = 2cos²(A/2) − 1:
   cos(A/2) = ±√[(1 + cos A)/2]

Dividing:
   tan(A/2) = ±√[(1 − cos A)/(1 + cos A)]
    = sin A / (1 + cos A)
    = (1 − cos A) / sin A

The ± sign depends on the quadrant of A/2. In a specific problem, the quadrant tells you which sign to pick.
⚠️

The ± Sign — A Source of Errors

The square root forms of sin(A/2) and cos(A/2) carry a ±. This sign is not arbitrary — it is determined by the quadrant in which A/2 lies. A common exam trap: the question specifies a range for A (e.g. π < A < 2π), which places A/2 in the second quadrant, making cos(A/2) negative. Always determine the quadrant of A/2 from the given constraint before applying the half-angle form. This step — identifying the sign — must be shown explicitly in a proof or solution.

The t-substitution (Weierstrass substitution)

Let t = tan(A/2). This single substitution converts every trig ratio of A into a rational function of t. It comes directly from sub-multiple forms and is a powerful tool in both integration and equation-solving:

t-Substitution: t = tan(A/2) sin A = 2t / (1 + t²)
cos A = (1 − t²) / (1 + t²)
tan A = 2t / (1 − t²)

Where these come from: substitute t into the double-angle formulas for tan to express sin A and cos A.
Application: solving equations of the form a sin A + b cos A = c by converting to a quadratic in t.
Problem Solving

A Practical Strategy for Any Multiple Angle Problem

Most students approach trig identity problems by staring at both sides until something clicks. That’s not a strategy — it’s hoping. Here’s a structured approach that works for the vast majority of exam questions in this topic.

Step 01

Identify what’s in the problem

Are you dealing with sin/cos/tan of 2A, 3A, or A/2? Are there products like sin A cos A? Powers like sin²A? Write down exactly what you see before doing anything.

Step 02

Decide: prove or simplify?

If proving an identity, pick the more complex side to start from. If simplifying, look for the substitution that reduces the complexity fastest. Don’t work on both sides simultaneously in a proof.

Step 03

Match the target form

Look at the other side (or the simplified form you’re aiming for). Which version of cos 2A gets you closer — the sin form, the cos form, or the mixed form? Choose the form that matches what you’re heading toward.

Step 04

Convert all ratios to the same function

If the target expression involves only sin, convert everything to sin. If it’s only cos, go that route. Mixed proofs often need converting via the Pythagorean identity. Consistent function choice prevents circular algebra.

Step 05

Use the Pythagorean identity freely

sin²A + cos²A = 1 is your most flexible tool. You can replace sin²A with 1 − cos²A and vice versa at any point. This is often what creates a chain of simplifications in a proof that looks stuck.

Step 06

Check that every line follows logically

Each step must follow from the previous one by a legitimate operation. Don’t assume the result midway through a proof — that’s circular reasoning. If you get stuck, try working backward from the target to find a bridge.

In trig proofs, the two most common mistakes are choosing the wrong form of cos 2A and multiplying both sides by something — which is solving an equation, not proving an identity. An identity proof must flow in one direction only.

— Standard pedagogy in A-Level and JEE trig instruction
Exam Readiness

Types of Exam Questions — What You’re Actually Being Asked to Do

Problems in this topic fall into a small number of recognisable categories. Knowing which category a question belongs to tells you immediately which approach to use.

Question Type What It Asks Core Approach Key Formula Family
Identity Proof Prove that LHS = RHS where both sides involve multiple/sub-multiple angles Start from the more complex side. Convert using double/triple angle formulas. Apply Pythagorean identity to match the simpler side. cos 2A (all 3 forms), sin 2A, tan 2A
Expression Simplification Simplify an expression involving sin 2A, cos 2A, etc. into a single trig ratio or constant Identify the substitution that collapses the expression. Often: recognise a product as (1/2) sin 2A, or a squared term as a cos 2A form. sin 2A = 2 sin A cos A; cos²A forms
Find the Value Given sin A = k or cos A = k (with quadrant), find sin 2A, cos 3A, tan(A/2), etc. Compute cos A (or sin A) from the Pythagorean identity first. Then substitute directly into the relevant multiple angle formula. All families — depends on the required angle multiple
Equation Solving Solve equations like sin 2A = cos A, or cos 2A + sin A = 0, over a given interval Convert everything to the same angle (usually A). Expand double angle. Factorise and solve. Watch for extraneous solutions from the domain restriction. Double angle substitutions; factorisation
Half-Angle Evaluation Evaluate sin 22.5°, cos 15°, tan(π/8), etc. exactly using sub-multiple formulas Identify the double of the target angle (45°, 30°, π/4). Use the exact value of the doubled angle in the half-angle formula. Determine the ± sign from the quadrant of the target angle. Half-angle explicit forms; exact values of standard angles
Integral Reduction Integrate sin²x, cos³x, sin 2x cos x, etc. using multiple angle substitutions Replace sin²x and cos²x using rearranged cos 2x forms. Replace sin³x or cos³x using the triple angle rearrangement. Then integrate standard forms. sin²A = (1 − cos 2A)/2; cos³A rearrangement

Strategy Builder — Approaching a Trig Identity Proof Correctly

What a well-structured proof looks like at each stage — with common errors shown for contrast

Choose Your Side
✓ Start from the more complex side and work toward the simpler one. State explicitly: “Taking the LHS:” then work down to “= RHS. Hence proved.” ✗ Work on both sides simultaneously toward the middle. This is circular and does not constitute a valid proof. In an exam, you lose marks for working from both ends even if the algebra is correct. A proof must be a one-directional chain of equalities.
First Substitution
✓ Identify the multiple angle expression (sin 2A, cos 3A, etc.) and immediately replace it with its expanded form. Show which formula you are applying — e.g. “Using sin 2A = 2 sin A cos A:” ✗ Substitute the wrong form of cos 2A (e.g. using cos²A − sin²A when the expression only has sin terms, then getting stuck). Naming the formula you’re using is not just good practice — it signals to the examiner that you know why you’re making each move, not just what move you’re making.
Algebraic Flow
✓ After expanding, combine like terms immediately. Don’t leave an expression with four terms when two of them cancel — resolve each line before moving to the next substitution. ✗ Accumulate expanding and substituting over multiple steps without simplifying, making errors harder to catch and the proof harder to follow. Clean algebra line by line makes errors visible before they compound. Messy algebra at this stage is where most marks are lost.
Conclusion
✓ The last line of your proof should be exactly equal to the other side of the identity. Write “= RHS” and follow with “Hence proved” or the QED symbol. Don’t leave the reader to infer the conclusion. In a 3–4 mark identity question, the conclusion line can carry a mark on its own. Always close the proof explicitly.
Your Problems

Approaching the Handwritten Problem Types — What the Image Shows

The uploaded image contains several handwritten trig expressions involving multiple angle forms. Without solving them for you, here’s how to approach each category that appears:

Product Expressions

Expressions like sin A · cos A or similar products

Whenever you see sin A times cos A in a problem, the first instinct should be: this is half of sin 2A. Write sin A cos A = (1/2) sin 2A and see if the expression collapses. This single recognition resolves a large fraction of simplification problems in this topic.

Squared Terms

Expressions involving sin²A or cos²A

Replace immediately using the rearranged cos 2A forms: sin²A = (1 − cos 2A)/2 and cos²A = (1 + cos 2A)/2. These conversions reduce “power-of-trig” expressions to first-degree trig expressions, which are far easier to combine or integrate.

Fraction Forms

Fractions with trig expressions in numerator and denominator

Look for factorisation opportunities. Expressions like (sin 2A) / (1 + cos 2A) often simplify using sub-multiple forms — because 1 + cos 2A = 2cos²A, turning the fraction into tan A. Recognise the standard forms before expanding everything.

Verify or Prove Format

Problems asking you to show one expression equals another

Choose the more complex side. Expand any double or triple angle expressions on that side. Apply the Pythagorean identity where you see sin² + cos² or sums/differences. Keep going until you reach the simpler side. If you’re stuck after two steps, try a different form of cos 2A — the right form usually unlocks the proof immediately.

Algebraic Identities with t

Problems involving expressions in tan(A/2)

If the problem has both sin A and cos A and you need to express the result in tan(A/2), use the t-substitution: sin A = 2t/(1+t²), cos A = (1−t²)/(1+t²) where t = tan(A/2). This converts the expression to a rational function, and simplification is then just algebra.

✅

General Checklist Before Starting Any Problem in This Topic

  • Have you identified the angle multiples present — is it 2A, 3A, or A/2?
  • Have you written down the relevant formula before doing any algebra?
  • If proving an identity — which side are you starting from, and why?
  • Have you checked the quadrant for any ± sign decision in half-angle forms?
  • Are all your trig ratios for the same angle before you start combining terms?
  • Have you checked whether sin A · cos A = (1/2)sin 2A applies?
Quality Control

Common Mistakes in Multiple Angle Problems — And What to Do Instead

#❌ MistakeWhy It Goes Wrong✓ Fix
1 Writing sin 2A = sin 2 · sin A or similar Treating the “2” as a coefficient outside the function. It is not. sin 2A means sine of the angle 2A — a completely different value from 2 sin A in general. Always write out “sin(2A) = 2 sin A cos A” in full. Never drop the compound structure.
2 Using the wrong form of cos 2A mid-proof and getting stuck The three forms of cos 2A are equivalent but not equally useful in every context. Choosing the form that doesn’t match the rest of the expression forces ugly algebra. Before substituting, look at the target expression and pick the cos 2A form that eliminates the trig function you don’t want in the final answer.
3 Working from both sides of a proof simultaneously This is not a valid proof technique. It looks like progress but doesn’t establish that LHS = RHS — you’ve only shown that if LHS = RHS then something equals itself. Pick one side. Work from it to the other. Only one direction. If you want to verify your approach, you can work backward privately — but your written proof must read one way.
4 Forgetting the ± in half-angle expressions sin(A/2) = ±√[(1 − cos A)/2]. Dropping the ± and always writing + is a sign error that changes the value of the expression for angles in the second half-period. Always determine the quadrant of A/2 from the given constraint on A. Then state explicitly which sign you’re using and why.
5 Expanding triple angle directly without using double angle as a bridge Attempting to “memorise” sin 3A = 3 sin A − 4 sin³A without knowing how to derive it means you can’t reconstruct it under pressure or adapt to related forms. Derive triple angle results every time by writing 3A = 2A + A and using the addition formula. The derivation takes about four lines and is faster than memorisation in exam conditions.
6 Stopping too early in a simplification Reaching an intermediate form like 2 sin A cos A and writing it as the final answer — without recognising this equals sin 2A — misses the point of the substitution. Always ask: can this expression be written as a single named function? Check whether products, sums, or squared terms match a known identity form.
7 Applying double angle formulas to sub-multiples incorrectly Writing sin(A/2) = 2 sin(A/4) cos(A/4) is correct but students sometimes write it as 2 sin A cos A — forgetting to halve the angle argument throughout. When applying double angle with the substitution A → A/2, every A in the formula must become A/2. Check all terms.
Further Reading

Sources and Resources for Mastering This Topic

📖

NCERT Mathematics Class 11 — Chapter 3

The standard Indian curriculum textbook covers compound angle formulas and derives double angle results explicitly. Chapter 3 (Trigonometric Functions) is the primary syllabus reference for this topic, including worked examples on proving identities and evaluating half-angle expressions.

ncert.nic.in · Free PDF available officially
🌐

Khan Academy — Trigonometric Identities

Khan Academy’s double angle formula unit provides free, step-by-step video derivations of sin 2A, cos 2A, and tan 2A, with practice problems graded by difficulty. A verified external resource used across secondary and undergraduate maths globally.

khanacademy.org/math/trigonometry
📚

S.L. Loney — Plane Trigonometry

The classic reference for competitive exam preparation. Chapters on compound angles, multiple angles, and submultiple angles contain hundreds of graded problems with full proofs. Standard recommended reading for JEE and olympiad-level trig.

Available through major academic booksellers
🎓

MIT OpenCourseWare — Single Variable Calculus

MIT’s free calculus materials show how double and half angle formulas are used in integration — specifically for integrating powers of trig functions. Provides the applied context that shows why these formulas matter beyond the trigonometry classroom.

ocw.mit.edu/courses/18-01
✏️

Smart Academic Writing — Maths Assignment Help

For students who need help structuring proofs, working through identity problems, or preparing maths assignments, our specialists cover Class 11–12 trig, JEE-level problems, and undergraduate calculus applications of multiple angle formulas.

Maths Help · Maths Tutoring
🔬

Paul’s Online Math Notes — Trig Formulas

A widely used free resource for university-level mathematics. The trig cheat sheet and worked examples for double and half angle substitutions are particularly useful for calculus students applying these identities in integration and differential equations contexts.

tutorial.math.lamar.edu

Need Help With Trig Identities and Proofs?

Our maths specialists can work through multiple angle formula problems with you — from Class 11 identity proofs to JEE-level questions and undergraduate calculus applications.

Get Expert Maths Help →
Common Questions

FAQs — Multiple Angle and Sub-Multiple Angle Formulas

What is the difference between multiple angle and sub-multiple angle formulas?
Multiple angle formulas express trig ratios of nA (where n ≥ 2) in terms of ratios of A. Sub-multiple angle formulas work in the opposite direction — they express ratios of A in terms of ratios of A/2. They are the same identities, just with the variable renamed. If you know the double angle formulas, you automatically know the sub-multiple forms: replace every A with A/2 and every 2A with A throughout the formula. That’s all sub-multiple angle formulas are. Many students treat them as a separate memorisation task. They’re not.
How do I know which form of cos 2A to use in a proof?
Look at what the target expression (the other side of the identity, or the simplified form you’re aiming for) contains. If the target has only sin terms, use 1 − 2sin²A — this leaves sin in the expression and eliminates cos. If the target has only cos terms, use 2cos²A − 1. If the target is mixed or symmetric, use cos²A − sin²A. If everything is in tan, use the tan form: (1 − tan²A)/(1 + tan²A). The right choice usually makes the next step obvious; the wrong choice leads to circular algebra.
How do you derive sin 3A without memorising the formula?
Write sin 3A = sin(2A + A). Apply the compound addition formula: sin(2A + A) = sin 2A · cos A + cos 2A · sin A. Now substitute sin 2A = 2 sin A cos A and cos 2A = 1 − 2sin²A. Expand: 2 sin A cos²A + (1 − 2sin²A) sin A. Replace cos²A = 1 − sin²A: 2 sin A(1 − sin²A) + sin A − 2sin³A = 2 sin A − 2sin³A + sin A − 2sin³A = 3 sin A − 4sin³A. That’s the complete derivation in under two minutes. No memorisation needed — just the addition formula and the double angle results you already know.
How do you evaluate sin 22.5° exactly using half-angle formulas?
22.5° is half of 45°. Use the half-angle formula: sin(A/2) = √[(1 − cos A)/2]. Here A = 45°, so cos 45° = 1/√2. Substitute: sin 22.5° = √[(1 − 1/√2)/2] = √[(√2 − 1)/(2√2)] = √[(√2 − 1)/2√2]. Rationalise to get the standard exact form. Since 22.5° is in the first quadrant, sin 22.5° is positive — the ± becomes + without any ambiguity here. This is the standard half-angle approach for evaluating any angle that is half of a standard angle (30°, 45°, 60°, 90°).
What is the t-substitution and when should I use it?
The t-substitution sets t = tan(A/2), which converts sin A = 2t/(1 + t²) and cos A = (1 − t²)/(1 + t²). Use it when: (a) you need to solve an equation of the form a sin A + b cos A = c (it converts this to a quadratic in t); (b) you need to integrate a rational expression in sin A and cos A (it converts the integral to a rational function of t, integrable by partial fractions); (c) you need to prove an identity that’s expressed in terms of A but is simplest in terms of A/2. It’s a powerful technique but introduce it only when the problem clearly involves both sin and cos and you need a single variable approach.
How are multiple angle formulas used in integration?
There are three main uses in calculus. First, to integrate sin²x or cos²x: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2 to reduce the power before integrating. Second, to integrate sin³x or cos³x: use the triple angle rearrangements sin³x = (3 sin x − sin 3x)/4 and cos³x = (3 cos x + cos 3x)/4. Third, to integrate products like sin mx · cos nx: use the product-to-sum formulas (which are rearrangements of compound angle formulas) to convert the product into a sum of first-degree trig terms. Each technique converts a “hard” integral into a standard one.
Can Smart Academic Writing help with trig assignments and proofs?
Yes. Our mathematics specialists are familiar with Class 11–12 trig, JEE/NEET question patterns, A-Level Further Maths, and undergraduate calculus. We can help you structure proofs correctly, work through identity problems step by step, check your algebra for errors, and prepare your assignments for submission. Visit our maths homework help or maths tutoring pages, or contact us directly.
Wrapping Up

You Don’t Need to Memorise 15 Formulas. You Need to Know 3 Things Well.

The compound addition formulas for sin and cos. The Pythagorean identity. And the move: set B = A.

Every double angle formula comes from the first of those with the third move applied. Every triple angle formula is the same process, one more time. Sub-multiple formulas are double angle formulas with A/2 substituted. And the t-substitution is just sub-multiple forms dressed up for algebra and calculus use.

The exam questions in this topic — identity proofs, simplifications, value-finding, equation-solving, half-angle evaluations — all reduce to recognising which formula family applies, choosing the right form, and keeping the algebra clean. The strategy is the same every time. The formulas change; the approach doesn’t.

If you’re working through problems and hitting walls, the issue is almost always one of two things: choosing the wrong form of cos 2A, or working algebraically in both directions simultaneously in a proof. Fix either of those and most problems resolve themselves.

For expert help working through specific problems, proof-checking, or maths assignment preparation, the team at Smart Academic Writing is available. See our maths homework help, online tutoring, and academic writing services.