$\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}} = ?$

If $A, B$ and $C$ are interior angles of a triangle $ABC$,then show that $\sin \left(\frac{B+C}{2}\right) = \cos \frac{A}{2}$.

Evaluate the following:
$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$(\operatorname{cosec} A - \sin A)(\sec A - \cos A) = \frac{1}{\tan A + \cot A}$

State whether the following is true or false. Justify your answer.
$\cot A$ is not defined for $A = 0^{\circ}$.

Show that:
$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ} = 1$
$(ii)$ $\cos 38^{\circ} \cos 52^{\circ} - \sin 38^{\circ} \sin 52^{\circ} = 0$