Double Angle Identities Explained with Examples

Double angle identities help you rewrite trig expressions that contain 2x. They are used in simplification, solving equations, calculus, graph transformations and exam questions where one trig form is much easier than another.

The key is not just memorising the formulas. The exam skill is choosing the version that fits the expression in front of you.

Quick Answer

What Double Angle Identities Are

Double angle identities are trigonometric identities that express a function of 2x using functions of x.

They are called identities because they are true for all values where both sides are defined. In exam work, they are most useful when they make an expression shorter or turn a difficult equation into a familiar one.

The Three Core Formulas

You should know:

sin 2x = 2 sin x cos x

cos 2x = cos^2 x - sin^2 x

tan 2x = 2 tan x / (1 - tan^2 x)

The cosine identity has two extra forms:

cos 2x = 1 - 2 sin^2 x

cos 2x = 2 cos^2 x - 1

These come from sin^2 x + cos^2 x = 1.

How to Choose the Right Formula

If the expression contains sin x cos x, try sin 2x.

If the expression contains sin^2 x, try cos 2x = 1 - 2 sin^2 x.

If the expression contains cos^2 x, try cos 2x = 2 cos^2 x - 1.

If the expression contains tan x, try tan 2x, but remember that the denominator cannot be zero.

When Double Angle Identities Appear in IB Maths AA

Double angle identities often appear when a question asks you to simplify a trigonometric expression, prove an identity, or solve an equation involving sin 2x, cos 2x or tan 2x.

Use them when you see:

• 2 sin x cos x
• 1 - 2 sin^2 x
• 2 cos^2 x - 1
• tan expressions that can become tan 2x

In IB Maths AA, the key is not just memorising the formulas, but choosing the form that makes the expression simpler.

Worked Example

Question: If sin x = 3/5 and cos x = 4/5, find sin 2x and cos 2x.

For sin 2x:

sin 2x = 2 sin x cos x

sin 2x = 2 x 3/5 x 4/5 = 24/25

For cos 2x:

cos 2x = cos^2 x - sin^2 x

cos 2x = (4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25

Common Mistakes

Mini Practice

1. Rewrite 2 sin x cos x.
2. Rewrite 1 - 2 sin^2 x.
3. Rewrite 2 cos^2 x - 1.
4. If tan x = 1/2, find tan 2x.
5. Explain why cos 2x has more than one useful form.

Answers:

1. sin 2x
2. cos 2x
3. cos 2x
4. tan 2x = 2(1/2) / (1 - 1/4) = 4/3
5. The different forms come from replacing sin^2 x or cos^2 x using sin^2 x + cos^2 x = 1.

Practice This Topic

Try this exam-style task:
Simplify 1 - 2 sin^2 x and explain why this form is useful in solving trig equations.

Answer guide:
1 - 2 sin^2 x is equal to cos 2x. This is useful because it changes a squared trig expression into a single trig function with angle 2x, which can make solving or graphing easier.

Practice related maths questions
Browse IB Maths AA SL practice
Some questions may require an EduNinja membership.

Browse broader IB practice
Browse the IB Question Bank

Related Study Links

• IB Maths AA Differentiation: Common Mistakes and Revision Guide
• IB Maths AA Integration: Area, Antiderivatives, and Common Mistakes
• Browse IB Maths AA SL practice
• Browse the IB Question Bank

FAQ

What are double angle identities?

Double angle identities are formulas that rewrite sin 2x, cos 2x and tan 2x using trig functions of x. They help simplify expressions and solve equations.

Which double angle identity should I use?

Choose the identity that matches the expression. Use sin 2x for 2 sin x cos x, use the sine or cosine version of cos 2x depending on the squared term, and use tan 2x for tan expressions.

Why does cos 2x have three forms?

The three forms are equivalent because sin^2 x + cos^2 x = 1. You can replace sin^2 x or cos^2 x to create the version that best fits the question.

Final Takeaway

Double angle identities are most useful when you choose the form that simplifies the question. Memorise the formulas, but practise the decision step too.