Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
(l mark)
Given that \(\cos x \neq 0\), show that the equation
$$\sin 2 x = \cos x \cos 2 x$$
can be written in the form
$$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.