State whether the following is true or false. Justify your answer.
$\sin (A+B) = \sin A + \sin B$

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

Evaluate:
$\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$

Evaluate:
$\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}$

Prove the following identity,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 the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$