Double angle formulae

The formulae

• sin 2A ≡ 2sin A cos A
• cos 2A ≡ cos² A - sin² A ≡ 2cos² A - 1 ≡ 1 - 2sin² A
• tan 2A ≡ 2tan A/(1 - tan² A)

Learn all three versions of cos 2A.

Choosing a version

Pick the version of cos 2x that matches the rest of the equation: if the equation also has sin x, use 1 - 2sin² x; if it has cos x, use 2cos² x - 1. Never divide by sin x or cos x: factorise instead, or you lose solutions.

Half angles

Replace A by A/2: sin A ≡ 2sin(A/2)cos(A/2) and cos A ≡ 2cos²(A/2) - 1 ≡ 1 - 2sin²(A/2). Rearranged versions, such as sin² x ≡ (1 - cos 2x)/2, are used for integration.

Worked example

Solve cos 2x = sin x for 0° ≤ x ≤ 360°.

1. Use cos 2x = 1 - 2sin² x: 2sin² x + sin x - 1 = 0.
2. (2sin x - 1)(sin x + 1) = 0
3. sin x = 1/2 gives 30°, 150°; sin x = -1 gives 270°.

Answer: x = 30°, 150° or 270°

Check yourself

1. x is acute and sin x = 3/5. Find sin 2x. Give your answer as a fraction in its simplest form.
   Show answer

   24/25. cos x = 4/5, so sin 2x = 2(3/5)(4/5) = 24/25.

2. cos x = 1/3. Find the exact value of cos 2x. Give your answer as a fraction in its simplest form.
   Show answer

   -7/9. cos 2x = 2cos² x - 1 = 2/9 - 1 = -7/9.

3. tan x = 2. Find tan 2x. Give your answer as a fraction in its simplest form.
   Show answer

   -4/3. tan 2x = 4/(1 - 4) = -4/3.

4. Solve sin 2x = sin x for 0° ≤ x ≤ 180°.
   Show answer

   x = 0°, 60° or 180°. 2sin x cos x - sin x = 0, so sin x(2cos x - 1) = 0. sin x = 0 gives 0°, 180°; cos x = 1/2 gives 60°.

5. You need to solve cos 2x + 3sin x = 2. Which version of cos 2x should you use?
   • 2cos² x - 1
   • cos² x - sin² x
   • 1 - 2sin² x
   • 2sin x cos x
   Show answer

   1 - 2sin² x. Using 1 - 2sin² x gives a quadratic in sin x only.

6. Which identity expresses cos x in terms of half angles?
   • cos x ≡ 2cos²(x/2) - 1
   • cos x ≡ 2cos(x/2)
   • cos x ≡ cos²(x/2) + sin²(x/2)
   • cos x ≡ 1 - 2cos²(x/2)
   Show answer

   cos x ≡ 2cos²(x/2) - 1. Put A = x/2 in cos 2A ≡ 2cos² A - 1.

7. Evaluate 2cos²(22.5°) - 1, to 3 decimal places.
   Show answer

   0.707. 2cos² A - 1 = cos 2A = cos 45° = sqrt(2)/2 = 0.707.

8. Simplify (1 - cos 2x)/sin 2x.
   • cot x
   • sin x
   • sec x
   • tan x
   Show answer

   tan x. 1 - cos 2x = 2sin² x and sin 2x = 2sin x cos x, so the fraction is sin x/cos x = tan x.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.