Question 7 - A-Level Maths

1. By first expanding $( \cos \theta + \sin \theta ) ^ { 2 }$, find the three solutions of the equation $$( \cos \theta + \sin \theta ) ^ { 2 } = 1$$ for $0 \leqslant \theta \leqslant \pi$.
2. Hence verify that the only solutions of the equation $\cos \theta + \sin \theta = 1$ for $0 \leqslant \theta \leqslant \pi$ are 0 and $\frac { 1 } { 2 } \pi$.
Prove the identity $\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } \equiv \frac { \cos \theta + \sin \theta - 1 } { 1 - 2 \sin ^ { 2 } \theta }$.
Using the results of (a) (ii) and (b), solve the equation $$\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } = 2 ( \cos \theta + \sin \theta - 1 )$$ for $0 \leqslant \theta \leqslant \pi$.