Evaluate:

$\operatorname{cosec} 31^{\circ}-\sec 59^{\circ}$

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

Evaluate:

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

If $\sin ( A - B )=\frac{1}{2}, \cos ( A + B )=\frac{1}{2}, 0^{\circ} < A + B \leq 90^{\circ}, A > B ,$ find $A$ and $B$

Given $15 \cot A =8,$ find $\sin A$ and $\sec A .$

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

$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$