NDA Maths · Trigonometric Identities

Double, Triple & Half-Angle

Double-angle (the most-used), triple-angle, and half-angle formulas — plus the symmetric tricks like sin α + cos α = p that feed straight into sin 2α.

All Trigonometric Identities notes NDA Maths strategy

Why this matters

Half of this subtopic's questions are HARD — the densest difficulty pocket in the chapter. Most resolve to picking the right form of cos 2A, knowing sin 3A = 3 sin A − 4 sin³A, or recognising a half-angle in 1 ± cos A.

Concept 1 of 4

Double-angle formulas

Intuition

Set B = A in the compound formulas. Cosine of a double angle has three interchangeable forms — the art is choosing the one that matches what you're given (sin only, cos only, or tan only).

Definition

$\sin 2A=2\sin A\cos A=\dfrac{2\tan A}{1+\tan^2 A}$ .
$\cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2 A-1=\dfrac{1-\tan^2 A}{1+\tan^2 A}$ .
$\tan 2A=\dfrac{2\tan A}{1-\tan^2 A}$ . Also $\tan A+\cot A=\dfrac{2}{\sin 2A}$ .

The three forms of cos 2A

\cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2 A-1

Worked example

\tan A=\tfrac34

, find

\sin 2A

$\sin 2A=\dfrac{2\tan A}{1+\tan^2 A}=\dfrac{2\cdot\tfrac34}{1+\tfrac{9}{16}}$ .
$=\dfrac{3/2}{25/16}=\dfrac{24}{25}$ .

Answer:

\sin 2A=\tfrac{24}{25}

Practice this conceptself-check · 4 quick reps

Try it yourself

With

\tan\alpha=\tfrac34

, find

2\sin 2\alpha+\cos 2\alpha

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\sin 2A=?$
2.
$\cos 2A$ in terms of $\sin A$ ?
3.
$\tan A+\cot A=?$
4.
$\dfrac{2\tan\theta}{1+\tan^2\theta}=?$

From the bank · past-year question

Example 1Trigonometric IdentitiesMODERATE

Let

2\sin\alpha + \cos\alpha = 2

, where

0 < \alpha < 90°

What is

2\sin 2\alpha + \cos 2\alpha

equal to?

[Q45 · Apr · 2025]

Drill 11 more on double-angle formulas

Concept 2 of 4

Triple-angle formulas

Intuition

The triple-angle identities turn sin 3A / cos 3A back into powers of sin A / cos A — and run in reverse to collapse "3 sin A − 4 sin³A" into a single sin 3A.

Definition

$\sin 3A=3\sin A-4\sin^3 A$ .
$\cos 3A=4\cos^3 A-3\cos A$ .
$\tan 3A=\dfrac{3\tan A-\tan^3 A}{1-3\tan^2 A}$ .

Triple angle

\sin 3A=3\sin A-4\sin^3 A,\qquad \cos 3A=4\cos^3 A-3\cos A

Worked example

Simplify

3\sin 20°-4\sin^3 20°

This is exactly $\sin 3A$ with $A=20°$ .
$=\sin 60°=\tfrac{\sqrt3}{2}$ .

Answer:

\tfrac{\sqrt3}{2}

Practice this conceptself-check · 4 quick reps

Try it yourself

Simplify

\sin 3x+\cos 3x+4\sin^3 x-3\sin x

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\sin 3A=?$
2.
$\cos 3A=?$
3.
$4\cos^3 10°-3\cos 10°=?$
4.
$\tan 3A=?$

From the bank · past-year question

Example 2Trigonometric IdentitiesMODERATE

What is

\sin 3x + \cos 3x + 4\sin^3 x - 3\sin x + 3\cos x - 4\cos^3 x

equal to?

[Q25 · Apr · 2020]

Drill 4 more on triple-angle formulas

Concept 3 of 4

Half-angle formulas and 1 ± cos A / 1 ± sin A

Intuition

Read the double-angle formulas backwards. The key recognitions:

1-\cos A=2\sin^2\tfrac A2

1+\cos A=2\cos^2\tfrac A2

, and

1\pm\sin A=(\sin\tfrac A2\pm\cos\tfrac A2)^2

. These convert square roots into clean half-angle expressions.

Definition

$1-\cos A=2\sin^2\tfrac A2$ , $\;1+\cos A=2\cos^2\tfrac A2$ .
$\tan\tfrac A2=\dfrac{\sin A}{1+\cos A}=\dfrac{1-\cos A}{\sin A}$ .
$\csc A+\cot A=\cot\tfrac A2$ , $\;\csc A-\cot A=\tan\tfrac A2$ .
$1\pm\sin A=\left(\sin\tfrac A2\pm\cos\tfrac A2\right)^2$ (mind the sign when taking the root).

Worked example

Simplify

\dfrac{1-\cos 2\theta}{\sin 2\theta}

$1-\cos 2\theta=2\sin^2\theta$ and $\sin 2\theta=2\sin\theta\cos\theta$ .
Ratio $=\dfrac{2\sin^2\theta}{2\sin\theta\cos\theta}=\tan\theta$ .

Answer:

\tan\theta

Practice this conceptself-check · 4 quick reps

Try it yourself

Find

\tan\!\left(\tfrac{3\pi}{8}\right)=\tan 67.5°

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$1-\cos A=?$
2.
$1+\cos A=?$
3.
$\csc A+\cot A=?$
4.
$\sqrt{\tfrac{1-\cos A}{1+\cos A}}=?$ (acute $A$ )

From the bank · past-year question

Example 3Trigonometric IdentitiesMODERATE

What is the value of

\tan\!\left(\frac{3\pi}{8}\right)

[Q48 · Sep · 2022]

Drill 6 more on half-angle formulas and 1 ± cos a / 1 ± sin a

Concept 4 of 4

Symmetric tricks: sin ± cos, power reduction, sₙ patterns

Intuition

A cluster of questions square a symmetric expression to expose a double angle (sin α + cos α squared gives 1 + sin 2α), reduce a fourth power to multiple angles, or chase patterns in

t_n=\sin^n\theta+\cos^n\theta

Definition

Square the sum: $(\sin\alpha+\cos\alpha)^2=1+\sin 2\alpha$ , $(\sin\alpha-\cos\alpha)^2=1-\sin 2\alpha$ .
Power reduction: $\cos^4 x=\dfrac{3+4\cos 2x+\cos 4x}{8}$ (and similarly for $\sin^4$ ).
** $x+\tfrac1x=2\cos\theta\Rightarrow x^n+\tfrac{1}{x^n}=2\cos n\theta$ ** (De Moivre flavour).

Worked example

\sin\alpha+\cos\alpha=p

, express

\sin 2\alpha

in terms of

p

Square: $p^2=\sin^2\alpha+2\sin\alpha\cos\alpha+\cos^2\alpha=1+\sin 2\alpha$ .
So $\sin 2\alpha=p^2-1$ .

Answer:

\sin 2\alpha=p^2-1

Practice this conceptself-check · 4 quick reps

Try it yourself

x+\tfrac1x=2\cos\theta

, find

x^2+\tfrac{1}{x^2}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$(\sin\alpha+\cos\alpha)^2=?$
2.
$\sin\alpha+\cos\alpha=p\Rightarrow\sin 2\alpha=?$
3.
$x+\tfrac1x=2\cos\theta\Rightarrow x^n+\tfrac{1}{x^n}=?$
4.
$(\sin\alpha-\cos\alpha)^2=?$

From the bank · past-year question

Example 4Trigonometric IdentitiesMODERATE

\sin\alpha + \cos\alpha = p

, then what is

\cos^2(2\alpha)

equal to?

[Q38 · Apr · 2019]

Drill 5 more on symmetric tricks: sin ± cos, power reduction, sₙ patterns

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Double-angle formulas
The three forms of cos 2A
$\cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2 A-1$
Triple-angle formulas
Triple angle
$\sin 3A=3\sin A-4\sin^3 A,\qquad \cos 3A=4\cos^3 A-3\cos A$

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Trigonometric IdentitiesEASY

What is

\frac{2\tan\theta}{1+\tan^2\theta}

equal to?

[Q36 · Sep · 2018]

Example 2Trigonometric IdentitiesMODERATE

Let

p=\dfrac{1}{3}-\dfrac{\tan 3x}{\tan x}

and

q=1-3\tan^2 x,\ 0<x<\pi,\ x\neq\dfrac{\pi}{2}

For how many values of

x

does

\frac{1}{p}

become zero?

[Q76 · Sep · 2023]

Example 3Trigonometric IdentitiesHARD

Suppose

\cos A

is given. If only one value of

\cos(A/2)

is possible, then A must be

[Q48 · Apr · 2018]

Example 4Trigonometric IdentitiesEASY

x+\dfrac{1}{x}=2\cos\theta

, then what is

x^3+\dfrac{1}{x^3}

equal to?

[Q48 · Sep · 2025]

Example 5Trigonometric IdentitiesHARD

Consider the following for the items that follow: Let

\sin?eta

be the GM of

\sin?lpha

and

\cos?lpha

;

\tan\gamma

be the AM of

\sin?lpha

and

\cos?lpha

What is the value of

\sec2\gamma

[Q32 · Apr · 2023]

Drill every past-year question on this subtopic

30 questions from the bank — paginated, with cart and Word-export support.

Drill the 30 questions

Drill every past-year question on this subtopic

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