Prove the identity cos θ / (tan θ(1 – sin θ)) = 1 + 1/sin θ

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

1. [3] (i) cos θ / (tan θ(1 – sin θ)) = cos θ / ( (sin θ/cos θ) (1 – sin θ) ) = cos² θ / (sin θ(1 – sin θ)); M1 Use of t = s÷c; M1 Replaces cos²θ with 1 – sin²θ to form f(sinθ); A1 AG. Ensure all ok. Must show difference of 2 squares.; = (1 – sin² θ) / (sin θ(1 – sin θ)) = (1 – sin θ)(1 + sin θ) / (sin θ(1 – sin θ)) = (1 + sin θ) / sin θ = 1/sin θ + 1
2. [3] (ii) cos θ / (tan θ(1 – sin θ)) = 4; M1 Linking up to obtain sinθ = k.; A1 A1√ co. √ 180° - 1st answer providing there are no other solutions in the range 0° to 360°.; → sinθ = 1/3 → θ = 19.5°, 160.5°