CAIE P1 2017 June — Question 3

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2017
SessionJune
TopicReciprocal Trig & Identities

3
  1. Prove the identity \(\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } \equiv \frac { 1 - \sin \theta } { 1 + \sin \theta }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } = \frac { 1 } { 2 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{8a3f8707-67a4-4069-aba5-7e9496cb1748-06_572_460_258_845} The diagram shows a circle with radius \(r \mathrm {~cm}\) and centre \(O\). Points \(A\) and \(B\) lie on the circle and \(A B C D\) is a rectangle. Angle \(A O B = 2 \theta\) radians and \(A D = r \mathrm {~cm}\).