Evaluate the following:

$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$

Evaluate the following:

$\sin 60^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 60^{\circ}$

If $\angle A$ and $\angle B$ are acute angles such that $\cos A =\cos B ,$ then show that $\angle A =\angle B$.

Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$

$\sin 2 A=2 \sin A$ is true when $A=$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$