Prove the identity cot²θ - tan²θ ≡ 4 cot 2θ cosec 2θ.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] Express the left hand side in terms of sine and cos. cos²θ/sin²θ - sin²θ/cos²θ
2. [M1] Combine to a single term and factorise the numerator. (cos²θ - sin²θ)(cos²θ + sin²θ)/(sin²θcos²θ)
3. [M1] Use correct double angle formulae in numerator and denominator. 4cos2θ/(sin²2θ)
4. [A1] Obtain 4 cot 2θ cosec 2θ from correct working. 4cos2θ/(sin2θsin2θ)
5. [M1*] Use the identity to obtain an expression in one trigonometric function. cot²2x = 5/4 or tan²2x = 4/5
6. [A1] Obtain tan² 2x = 4/5. OE
7. [DM1] Obtain one solution e.g. 20.9°. AWRT
8. [A1] Obtain a second value e.g. 69.1° and no extras in range. AWRT