Question 8 - A-Level Maths

Express $\cos \theta + \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving the exact values of $R$ and $\alpha$.
Hence show that $$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
By differentiating $\frac { \sin x } { \cos x }$, show that if $y = \tan x$ then $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$.
Using the results of parts (ii) and (iii), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$