Build identity fluency first
Before you tackle proofs, drill the basic identities until you can simplify any short expression in under a minute.
Treat reductions as conversions
Every reduction formula converts an angle in another quadrant to an equivalent first-quadrant angle. That's all they do.
Reduction formulas, worked
Simplify sin(180° − x)·cos(−x) / sin(360° − x). sin(180° − x) = sin x, cos(−x) = cos x and sin(360° − x) = −sin x, so the expression is sin x cos x / (−sin x) = −cos x.
Always write the +k·360°
No period term = lost mark on general solutions.
Know which identities you must learn by heart
The information sheet gives the compound-angle and double-angle formulas and the sine, cosine and area rules. It does not give tanθ = sinθ/cosθ, sin²θ + cos²θ = 1 or the reduction formulas, so those must be memorised.
Compound angles without a calculator
Write the angle as a sum or difference of special angles. For example, sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(½) = (√6 + √2)/4.
A worked identity proof
Prove that (1 − cos²x)/(sinx·cosx) = tanx. Left side: sin²x/(sinx·cosx) = sinx/cosx = tanx, which is the right side. Work on one side at a time and never move terms across the equals sign in a proof.
A general solution in four steps
Solve 2sinx = −1.
- Isolate the ratio: sinx = −½.
- Reference angle: 30°.
- sin is negative in quadrants III and IV: x = 210° or x = 330°.
- Add the period: x = 210° + k·360° or x = 330° + k·360°, k ∈ ℤ.
3D problems, worked
From a point A on level ground, the angle of elevation to the top T of a vertical tower BT is 30°, and AB = 40 m. In right-angled △ABT, BT = 40 tan 30° ≈ 23.09 m. In a full 3D question, a second triangle on the ground, solved with the sine or cosine rule, usually gives you AB first.
Draw 3D problems flat
Pull out each triangle on its own. When a question stops you, snap it into Snap&Learn — you'll get a CAPS-aligned step-by-step explanation in seconds, and your first three AI solutions are free.