Double-Angle Identities Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Double-Angle Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Formulas expressing $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ in terms of single-angle trig functions.

What if both angles in the sum formula are the same? Setting $A = B = \theta$ in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $\sin 2\theta$ , $\cos 2\theta$ , $\tan 2\theta$ written from single-angle values, with cosine offering three interchangeable forms.

Common stuck point: The procedure for double-angle identities is the easy part; the trap is writing $\sin 2\theta=2\sin\theta$ . Asking "Is the angle exactly twice another, so I can express it from single-angle trig values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the angle exactly twice another, so I can express it from single-angle trig values?

Worked Examples

Example 1

easy

\cos(\theta) = \frac{3}{5}

, find

\cos(2\theta)

using the double-angle formula.

Answer

\cos (2 θ) = - \frac{7}{25}

First step

Use the double-angle formula:

\cos(2\theta) = 2\cos^2(\theta) - 1

Full solution

2
Substitute: $\cos(2\theta) = 2\left(\frac{3}{5}\right)^2 - 1 = 2 \cdot \frac{9}{25} - 1$ .
3
$= \frac{18}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}$ .

The double-angle formula for cosine has three equivalent forms:

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

. Choose the form that uses the information you have — here we used

2\cos^2\theta - 1

since we knew cosine.

Example 2

medium

Find

\sin(2\theta)

given that

\tan(\theta) = \frac{5}{12}

and

\theta

is in Quadrant I.

Example 3

medium

Solve

\cos(2x) = \cos x

[0, 2\pi)

Example 4

medium

Use power-reduction to write

\cos^2 x

in terms of

\cos(2x)

Example 5

hard

Prove that

\dfrac{\sin(2\theta)}{1 + \cos(2\theta)} = \tan\theta

Example 6

hard

Show that

\sin(3\theta) = 3\sin\theta - 4\sin^3\theta

using

\sin(3\theta) = \sin(2\theta + \theta)

Example 7

challenge

Find the maximum value of

f(\theta) = \sin\theta\cos\theta + \cos(2\theta)

[0, 2\pi)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Simplify

\cos^2(x) - \sin^2(x)

using a double-angle identity.

Example 2

hard

Solve

\sin(2x) = \cos(x)

for

x \in [0, 2\pi)

Example 3

easy

State the double-angle formula for

\sin(2\theta)

Example 4

easy

State one form of the double-angle formula for

\cos(2\theta)

Example 5

easy

Give the form of

\cos(2\theta)

in terms of

\cos^2\theta

only.

Example 6

easy

Give the form of

\cos(2\theta)

in terms of

\sin^2\theta

only.

Example 7

easy

\sin\theta = \frac35

and

\cos\theta = \frac45

, find

\sin(2\theta)

Example 8

easy

State the double-angle formula for

\tan(2\theta)

Example 9

easy

\cos\theta = \frac35

, use a double-angle form to find

\cos(2\theta)

Example 10

easy

\sin(2\theta) = 2\sin\theta

correct? Answer yes or no.

Example 11

medium

\sin\theta = \frac{5}{13}

and

\theta

is in Quadrant I, find

\cos(2\theta)

Example 12

medium

\cos\theta = -\frac35

and

\theta

is in Quadrant II, find

\sin(2\theta)

Example 13

medium

Simplify

2\sin 15°\cos 15°

Example 14

medium

Simplify

\cos^2 25° - \sin^2 25°

Example 15

medium

Simplify

1 - 2\sin^2 30°

Example 16

medium

\tan\theta = \frac13

, find

\tan(2\theta)

Example 17

medium

Express

\sin(4\theta)

using a double-angle formula once.

Example 18

medium

\sin\theta = \frac{4}{5}

and

\theta

is in Quadrant I, find

\cos(2\theta)

Example 19

medium

Simplify

\frac{2\tan 22.5°}{1 - \tan^2 22.5°}

Example 20

challenge

\cos\theta = \frac45

and

\theta

is in Quadrant IV, find

\sin(2\theta)

and

\cos(2\theta)

Example 21

challenge

Prove the power-reduction formula

\sin^2\theta = \frac{1 - \cos(2\theta)}{2}

Example 22

challenge

\sin(2\theta) = \frac12

, find the value of

\sin\theta\cos\theta

Example 23

easy

Compute

2\sin 45°\cos 45°

Example 24

easy

\sin\theta = \tfrac{1}{2}

and

\cos\theta = \tfrac{\sqrt{3}}{2}

, find

\sin(2\theta)

Example 25

easy

Simplify

2\cos^2 60° - 1

Example 26

easy

Find

\tan(2 \cdot 0°)

and use the formula to verify it equals

0

Example 27

easy

Simplify

1 - 2\sin^2 45°

Example 28

medium

\sin\theta = -\tfrac{3}{5}

and

\theta

is in Quadrant III, find

\sin(2\theta)

Example 29

medium

\cos\theta = \tfrac{7}{25}

and

\theta

is in Quadrant IV, find

\cos(2\theta)

Example 30

medium

Simplify

\dfrac{\sin(2\theta)}{\sin\theta}

Example 31

medium

Write

\cos 4\theta

in terms of

\cos 2\theta

using a double-angle formula.

Example 32

medium

\tan\theta = -\tfrac{1}{2}

, find

\tan(2\theta)

Example 33

medium

Express

\sin\theta\cos\theta

in terms of

\sin(2\theta)

Example 34

medium

Simplify

\dfrac{1 - \cos(2x)}{\sin(2x)}

Example 35

medium

\cos\theta = \tfrac{12}{13}

and

\theta

is in Quadrant I, find

\tan(2\theta)

Example 36

hard

Solve

\sin(2x) + \sin x = 0

[0, 2\pi)

Example 37

hard

\sin\theta + \cos\theta = \tfrac{1}{2}

, find

\sin(2\theta)

Example 38

hard

Solve

\cos(2x) = \sin x

[0, 2\pi)

Example 39

hard

Find the exact value of

\sin 22.5°

using a half-angle approach derived from the cosine double-angle identity.

Example 40

challenge

\sin(2\theta) = \tfrac{3}{5}

and

\theta \in (0, \pi/4)

, find

\sin\theta - \cos\theta

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Related Concepts

Sum and Difference Identities

Background Knowledge

These ideas may be useful before you work through the harder examples.

trig identities sum difference