8. (a) Prove that
$$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$
(b) Hence solve, for \(0 \leqslant \theta < \frac { \pi } { 2 }\)
- \(\sin 2 \theta - \tan \theta = \sqrt { 3 } \cos 2 \theta\)
- \(\tan ( \theta + 1 ) \cos ( 2 \theta + 2 ) - \sin ( 2 \theta + 2 ) = 2\)
Give your answers in radians to 3 significant figures, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)