Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$

(N/A) To prove: $\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
$L.H.S. = \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}$
Taking the common denominator:
$= \frac{\cos^2 A + (1+\sin A)^2}{(1+\sin A)(\cos A)}$
Expanding the numerator:
$= \frac{\cos^2 A + 1 + \sin^2 A + 2\sin A}{(1+\sin A)(\cos A)}$
Using the identity $\sin^2 A + \cos^2 A = 1$:
$= \frac{(\sin^2 A + \cos^2 A) + 1 + 2\sin A}{(1+\sin A)(\cos A)}$
$= \frac{1 + 1 + 2\sin A}{(1+\sin A)(\cos A)}$
$= \frac{2 + 2\sin A}{(1+\sin A)(\cos A)}$
Factoring out $2$:
$= \frac{2(1+\sin A)}{(1+\sin A)(\cos A)}$
$= \frac{2}{\cos A} = 2 \sec A$
$= R.H.S.$