Question

Prove the following statements.
$\frac{1}{cosecA-\cot A}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{cosecA+\cot A}$.

Answer 1

We have,

$LHS=$$\frac{1}{cosecA-\cot A}-\frac{1}{\sin A}$

$=\frac{1}{(cosecA-\cot A)}\times \frac{(cosecA+\cot A)}{(cosecA+\cot A)}-\frac{1}{\sin A}$

$=\frac{cosecA+\cot A}{(cose{c}^{2}A-{\cot }^{2}A)}-cosecA$

$=cosecA+\cot A-cosecA......................[\because cose{c}^{2}A-co{t}^{2}A=1]$

$=\cot A$

and

$RHS=$$\frac{1}{\sin A}-\frac{1}{cosecA+\cot A}$

$=\frac{1}{\sin A}-\frac{cosecA-\cot A}{(cosecA+\cot A)(cosecA+\cot A)}$

$=\frac{1}{sinA}-\frac{cosecA-\cot A}{cose{c}^{2}A+{\cot }^{2}A}$

$=cosecA-(cosecA-\cot A)[\because cose{c}^{2}A-{\cot }^{2}A=1]$

$=\cot A$

$LHS=RHS$