CAIE A-Level Mathematics A2 3.3.2 Trig Identities Questions

Use the double-angle formula:
$$\tan2x=\frac{2\tan x}{1-\tan^2x}.$$
For example,
$$\tan^3x+\frac{4\tan x}{1-\tan^2x}-\tan x=0.$$
Multiplying by $1-\tan^2x$ gives a correct equation in $\tan x$, such as
$$\tan^3x-\tan^5x+4\tan x-\tan x+\tan^3x=0.$$
Since $\tan x\ne0$, reduce correctly to
$$\tan^4x-2\tan^2x-3=0.$$

Using part (a) with $x=2\theta$:
$$\tan^4 2\theta-2\tan^2 2\theta-3=0.$$
Thus
$$\tan^2 2\theta=3,\qquad \tan2\theta=\pm\sqrt3.$$
For $0<\theta<\pi$,
$$\theta=\frac{\pi}{6},\quad \frac{\pi}{3},\quad \frac{2\pi}{3},\quad \frac{5\pi}{6}.$$
Exact answers only;
no others in the interval.