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Learn Index · Trigonometry Grade 11–12

Trigonometric identities

Equations true for all angles, such as sin²θ + cos²θ = 1.

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Always true

A trigonometric identity is true for every angle where both sides are defined. (A trig equation, by contrast, is true only for certain angles.)

Fundamental identities

Type Identity
Quotient identity tan θ = sin θ / cos θ
Reciprocal identities csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ
Pythagorean identity sin²θ + cos²θ = 1

The Pythagorean identity comes from the unit circle: x² + y² = 1 with x = cos θ, y = sin θ.

Rearranged forms:

  • sin²θ = 1 − cos²θ
  • cos²θ = 1 − sin²θ
  • 1 + tan²θ = sec²θ (divide by cos²θ)
  • 1 + cot²θ = csc²θ (divide by sin²θ)

Sum, difference and double angle formulas

Formula
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Example: Exact value of cos 15° using cos(45° − 30°):

cos 45 cos 30 + sin 45 sin 30 = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

How to prove an identity

  1. Work on one side (usually the more complicated one).
  2. Rewrite everything in terms of sin and cos.
  3. Use algebra: common denominators, factoring, expanding.
  4. Substitute identities.
  5. Stop when it matches the other side.

Don’t move terms across the equals sign: you’re showing equivalence, not solving.

Example proof 1

Prove: tan θ cos θ = sin θ

LS = (sin θ / cos θ)(cos θ) = sin θ = RS ✓

Example proof 2

Prove: (1 − cos²θ)/sin θ = sin θ

LS = sin²θ / sin θ = sin θ = RS ✓

Example proof 3

Prove: 1/(1 − sin θ) + 1/(1 + sin θ) = 2 sec²θ

LS = [(1 + sin θ) + (1 − sin θ)] / (1 − sin²θ) = 2/cos²θ = 2 sec²θ = RS ✓

Testing with a value

Substitute an angle (like 30°) into both sides. If they differ, it’s not an identity. If they match, it might be; you still need a proof.

Worked examples

Example 1: Simplify sin²x + cos²x + tan²x → 1 + tan²x = sec²x.

Example 2: Simplify (sin 2θ)/(2 cos θ) → sin θ.

Example 3: Exact value of sin 75° = sin(45 + 30) = (√6 + √2)/4.

Common mistakes

  • sin(A + B) = sin A + sin B: false.
  • Working on both sides at once as if solving an equation.
  • Dividing by expressions that might be zero.

Try it yourself

  1. Simplify cos θ tan θ.
  2. Simplify 1 − sin²x.
  3. Prove sec θ − cos θ = sin θ tan θ.
  4. Exact value of cos 75°.

Answers: 1) sin θ 2) cos²x 3) LS = 1/cos θ − cos θ = (1 − cos²θ)/cos θ = sin²θ/cos θ = sin θ tan θ ✓ 4) (√6 − √2)/4

Quick recap

  • Trigonometric identities are true for all allowed angles.
  • Key ones: quotient identity, reciprocal identities, Pythagorean identity, double angle formulas.
  • To prove, transform one side into the other.