4.(a)Prove the identity

$$( \sin x + \cos y ) \cos ( x - y ) \equiv ( 1 + \sin ( x - y ) ) ( \cos x + \sin y )$$

(b)Hence,or otherwise,show that

$$\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } \equiv \frac { 1 + \tan \theta } { 1 - \tan \theta }$$

(c)Given that $k > 1$ ,show that the equation $\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } = k$ has a unique solution in the interval $0 < \theta < \frac { \pi } { 4 }$

\hfill \mbox{\textit{Edexcel AEA 2019 Q4 [17]}}