Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable (where both sides are defined). At A-Level (9MA0), you must know the fundamental identities, use them to simplify expressions, prove further identities, and solve equations.

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The Two Fundamental Identities

Identity 1: tan(theta) = sin(theta) / cos(theta)

This follows directly from the definitions. It is valid for all theta where cos(theta) ≠ 0.

Identity 2: sin²(theta) + cos²(theta) = 1

This is the Pythagorean identity. It holds for all values of theta.

Rearrangements:

• sin²(theta) = 1 - cos²(theta)
• cos²(theta) = 1 - sin²(theta)

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Proving Identities

When proving a trigonometric identity, work with one side (usually the more complicated side) and show it simplifies to the other.

Golden rules:

1. Start with one side only — do NOT work both sides simultaneously.
2. Replace tan with sin/cos when useful.
3. Use sin² + cos² = 1 to eliminate one trig function.
4. Factorise where possible.

Worked Example 1

Prove that (1 - cos²x) / sin(x) = sin(x).

Start with the left side:

LHS = (1 - cos²x) / sin(x)

Since 1 - cos²x = sin²x: