Question

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

Answer 1

First we consider Left Hand Side
$\text{LHS}=\frac{\cot A+\text{cosec}A-1}{\cot A-\text{cosec}A+1}$

$=\frac{\cot A+\text{cosec}A-({\text{cosec}}^{2}A-{\cot }^{2}A)}{\cot A-\text{cosec}A+1}$

$=\frac{(\cot A+\text{cosec}A)-(\text{cosec}A+\cot A)(\text{cosec}A-\cot A)}{\cot A-\text{cosec}A+1}$

$=\frac{(\cot A+\text{cosec}A)(1-(\text{cosec}A-\cot A\left)\right)}{\cot A-\text{cosec}A+1}$

$=\frac{(\cot A+\text{cosec}A)(1-\text{cosec}A+\cot A)}{\cot A-\text{cosec}A+1}$

$=\cot A+\text{cosec}A$

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

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

Then, Right Hand Side,
$\text{RHS}=\frac{\cos A+1}{\sin A}$
Therefore, LHS = RHS
Hence Proved.