Trigonometric identities, explained with examples

By The STEM-Tutor team · Updated October 2026

Trig identities are equations that are true for every value of the angle. At A level you use them to simplify expressions, prove results, solve equations and integrate.

The core identities

• tan θ = sin θ / cos θ
• sin²θ + cos²θ = 1

The second comes straight from Pythagoras: on the unit circle, a point at angle θ has coordinates (cos θ, sin θ), and its distance from the origin is 1. Both identities are in the first year of A level Maths and you must know them from memory.

Reciprocal functions and Pythagorean identities

In the second year you meet three reciprocal functions: sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ = cos θ/sin θ. A useful memory aid: the third letter of each name tells you its partner (sec goes with cos, cosec with sin, cot with tan).

Dividing sin²θ + cos²θ = 1 by cos²θ and by sin²θ gives two more identities:

• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

You do not need to memorise these if you can derive them in a few seconds, which is a good habit.

Compound angle formulae

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B (note the sign swaps)
• tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

These are in the formula booklet for most boards, but you should be fluent at using them.

Double angle formulae

Set B = A in the compound angle formulae:

• sin 2A = 2 sin A cos A
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
• tan 2A = 2 tan A/(1 − tan²A)

The three forms of cos 2A are especially useful. Choose the one that matches the rest of the expression, so the equation ends up in a single trig function.

Worked examples

Example 1: an exact value

Find the exact value of sin 75°.

• sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°.
• = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Example 2: proving an identity

Prove that (1 − cos 2x)/sin 2x ≡ tan x.

• Start with the left-hand side. Use cos 2x = 1 − 2sin²x and sin 2x = 2 sin x cos x.
• LHS = (1 − 1 + 2sin²x)/(2 sin x cos x) = 2sin²x/(2 sin x cos x) = sin x/cos x = tan x = RHS. ∎

Example 3: proving with reciprocal functions

Prove that sec²x − tan²x ≡ 1, and that (1 + cos x)(cosec x − cot x) ≡ sin x.

• The first is 1 + tan²x = sec²x rearranged.
• Second: cosec x − cot x = (1 − cos x)/sin x. Multiply by (1 + cos x): (1 − cos²x)/sin x = sin²x/sin x = sin x. ∎

Example 4: solving with sin² + cos² = 1

Solve 2cos²x + 3 sin x = 3 for 0° ≤ x < 360°.

• Replace cos²x with 1 − sin²x: 2 − 2sin²x + 3 sin x = 3.
• Rearrange: 2sin²x − 3 sin x + 1 = 0, so (2 sin x − 1)(sin x − 1) = 0.
• sin x = 1/2 gives x = 30°, 150°. sin x = 1 gives x = 90°. x = 30°, 90°, 150°.

Example 5: solving with a double angle

Solve sin 2x = sin x for 0 ≤ x < 2π.

• 2 sin x cos x = sin x, so sin x(2 cos x − 1) = 0. Do not divide by sin x, or you lose solutions.
• sin x = 0: x = 0, π. cos x = 1/2: x = π/3, 5π/3.
• x = 0, π/3, π, 5π/3.

Example 6: R cos(x − α) form

Write 3 cos x + 4 sin x in the form R cos(x − α), and find its maximum value.

• R cos(x − α) = R cos x cos α + R sin x sin α. Compare: R cos α = 3 and R sin α = 4.
• R = √(9 + 16) = 5 and tan α = 4/3, so α ≈ 53.1°.
• 3 cos x + 4 sin x = 5 cos(x − 53.1°), with maximum value 5 when x = 53.1°.

Example 7: identities in integration

Find ∫ cos²x dx.

• Rearrange cos 2x = 2cos²x − 1 to cos²x = ½(1 + cos 2x).
• ∫ ½(1 + cos 2x) dx = x/2 + (sin 2x)/4 + C.

How to prove an identity

• Start from one side, usually the more complicated one, and work towards the other. Do not treat the identity as an equation and move terms across.
• Convert sec, cosec, cot and tan into sin and cos if you are stuck.
• Look for sin² + cos², double angles and fractions you can combine over a common denominator.
• Use the ≡ symbol and finish with a clear statement such as “= RHS”.

Common mistakes

• Dividing an equation by sin x or cos x and losing solutions where they equal zero.
• Writing sin 2x = 2 sin x, which is false.
• Getting the sign wrong in cos(A + B).
• Missing solutions in the range; sketch the graph or use the CAST diagram to find all of them.
• Using degrees when the range is in radians, or the other way round.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Which trig identities are in the A level formula booklet?

Most boards give the compound angle formulae. The basic identities sin² + cos² = 1 and tan = sin/cos, and the double angle formulae, usually need to be learned. Check your own board's booklet.

Are trig identities in AP Calculus?

AP Calculus uses the basic identities and double angle formulae, especially in integration, but it does not test proofs of identities the way A level does.

Key takeaways

• sin²θ + cos²θ = 1 and tanθ = sinθ/cosθ are the foundations.
• Divide by cos² or sin² to get the sec and cosec identities.
• Choose the form of cos 2A that leaves one trig function.
• Never divide by sin x or cos x when solving; factorise instead.
• To prove identities, work from one side to the other.

Related guides

• A level Maths formula booklet
• A level Maths integration
• GCSE Maths trigonometry