Question

Prove the following identity:
$\left(i\right)\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}=2\text{cosec}A$

Answer 1

First we consider Left Hand Side
$\text{LHS}=\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}$

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

We know that, $(a+b{)}^2=a^2+b^2+2ab$
Then,
$=\frac{{\sin }^{2}A+(1+{\cos }^{2}A+2\cos A)}{\sin A(1+\cos A)}$

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

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

$=\frac{2}{\sin A}$

$=2\text{cosec}A$
Now, Right Hand Side, $\text{RHS}=2\text{cosec}A$
Therefore RHS = LHS
Hence Proved.