Question

PROVE THAT
1/cosecA-cotA = cosecA+cotA​

Answer 1

Correct Question :-

$\mapsto \tt \: \frac{1}{cosecA + cotA} = cosecA - cotA$

Solution :-

▶LHS :-

$\mapsto \tt \: \frac{1}{cosecA + cotA}$

$\mapsto \tt \: \frac{1}{ \frac{1}{sinA} + \frac{cosA}{sinA} }$

$\mapsto \tt \: \frac{1}{ \frac{1 + cosA}{sinA} }$

$\mapsto \tt \: \frac{sinA}{(1 + cosA)}$

$\mapsto \tt \: \frac{sinA}{(1 + cosA)} \times \frac{(1 - cosA)}{(1 - cosA)}$

$\mapsto \tt \frac{sinA(1 - cosA)}{1 {}^{2} - cos {}^{2} A }$

$\mapsto \tt \: \frac{ \cancel{sinA}(1 - cosA)}{sin {}^{ \cancel2} A}$

$\mapsto \tt \: \frac{1 - cosA}{sinA}$

$\mapsto \tt \: \frac{1}{sin} - \frac{cosA}{sinA}$

$\mapsto \tt \: cosecA - cotA$

▶RHS.

Hence,

${ \boxed{ \tt \red {\frac{1}{cosecA + cotA} = cosecA - cotA}}}$