(a) Prove the identity \(\left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} + 1 \right) \equiv \frac{1}{\tan x}\).

(b) Hence solve the equation \(\left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} + 1 \right) = 2 \tan^2 x\) for \(0^\circ \leq x \leq 180^\circ\).