**(a)** $$LHS=\frac{\frac{\cos A}{\sin A}-\cos A}{\frac{\cos A}{\sin A}+\cos A}=\frac{\cos A\left(\frac{1}{\sin A}-1\right)}{\cos A\left(\frac{1}{\sin A}+1\right)}=\frac{1-\sin A}{1+\sin A}$$
Also, $\sec A-\tan A=\frac{1-\sin A}{\cos A}$ and $\sec A+\tan A=\frac{1+\sin A}{\cos A}$, so $$RHS=\frac{(1-\sin A)/\cos A}{(1+\sin A)/\cos A}=\frac{1-\sin A}{1+\sin A}$$ Hence LHS = RHS.

**(b)** $$LHS=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)=\frac{1-\sin^2A}{\sin A}\times\frac{1-\cos^2A}{\cos A}=\frac{\cos^2A}{\sin A}\times\frac{\sin^2A}{\cos A}=\sin A\cos A$$ $$RHS=\frac{1}{\tan A+\cot A}=\frac{1}{\frac{\sin^2A+\cos^2A}{\sin A\cos A}}=\sin A\cos A$$ Hence LHS = RHS.