Prove that sin²θ + cos²θ = 1. Hence prove that 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

1. 1 · Write what is given and what is to be found Given: right triangle ABC with ∠B = 90° and ∠A = θ. To prove: the three identities.
2. 2 · Draw the figure Right triangle ABC, right angle at B, θ at A. Label the opposite side BC, the adjacent side AB and the hypotenuse AC.
3. 3 · Write the theorem By Pythagoras theorem, AB² + BC² = AC² ... (1)
4. 4 · Divide (1) by AC² (AB/AC)² + (BC/AC)² = 1. Since AB/AC = cos θ and BC/AC = sin θ, cos²θ + sin²θ = 1.
5. 5 · Divide (1) by AB² 1 + (BC/AB)² = (AC/AB)². Since BC/AB = tan θ and AC/AB = sec θ, 1 + tan²θ = sec²θ.
6. 6 · Divide (1) by BC² (AB/BC)² + 1 = (AC/BC)². Since AB/BC = cot θ and AC/BC = cosec θ, cot²θ + 1 = cosec²θ.
7. 7 · Check and box the answer Box the three identities. Hence proved.

Scoring tip: Name each ratio (for example AB/AC = cos θ) before you replace it. That line shows the examiner you understand the proof.