Question

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

Answer 1

Consider the $LHS$ of the expression.

$\Rightarrow \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}$

$\Rightarrow \frac{{\cos }^{2}A+{(1+\sin A)}^2}{\cos A(1+\sin A)}$

$\Rightarrow \frac{{\sin }^{2}A+{\cos }^{2}A+1+2\sin A}{\cos A(1+\sin A)}$

$\Rightarrow \frac{2+2\sin A}{\cos A(1+\sin A)}$

$\Rightarrow \frac{2(1+\sin A)}{\cos A(1+\sin A)}$

$\Rightarrow \frac{2}{\cos A}$

$\Rightarrow 2\sec A=RHS$

Hence, proved.