25 (a) Prove that sqrt((1+ A)/(1− A)) = A + A

$$\text{LHS} = \sqrt{\frac{1+\sin A}{1-\sin A} \times \frac{1+\sin A}{1+\sin A}}$$ $$\text{LHS} = \sqrt{\frac{(1+\sin A)^2}{1^2 - \sin^2 A}}$$

$$\text{LHS} = \frac{\sqrt{(1+\sin A)^2}}{\sqrt{\cos^2 A}}$$ $$\text{LHS} = \frac{1+\sin A}{\cos A}$$

$$\text{LHS} = \sec A + \tan A$$

This is equal to the Right Hand Side (RHS).

$$\text{LHS} = \text{RHS}$$

Hence, proved.
$\boxed{\text{Proved}}$

25 (b) Evaluate $\frac{3 \cos^2 30^\circ - 6 \operatorname{cosec}^2 30^\circ}{\tan^2 60^\circ}$

$$\cos 30^\circ = \frac{\sqrt{3}}{2}$$ $$\operatorname{cosec} 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{1/2} = 2$$ $$\tan 60^\circ = \sqrt{3}$$

$$\frac{3 \left(\frac{3}{4}\right) - 6 (4)}{3}$$ $$\frac{\frac{9}{4} - 24}{3}$$

$$\frac{\frac{9}{4} - \frac{96}{4}}{3}$$ $$\frac{-\frac{87}{4}}{3}$$

$$-\frac{87}{4} \times \frac{1}{3}$$ $$-\frac{87}{12}$$

$$-\frac{29}{4}$$

The evaluated value is $\boxed{-\frac{29}{4}}$.