Evaluate the following:
$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$
State whether the following are true or false. Justify your answer.
$\sin (A+B)=\sin A+\sin B$
If $A , B$ and $C$ are interior angles of a triangle $ABC ,$ then show that
$\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$
Prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$