Question

Prove that : -
$cosec\theta +\cot \theta =\frac{1}{cosec\theta -\cot \theta }$

Answer 1

$cosec\theta +cot\theta =\frac{1}{cosec\theta -cot\theta }$

solving LHS.

$cosec\theta +cot\theta$ = (on rationalizing)

$\frac{(cosec\theta +cot\theta )(cosec\theta -cot\theta )}{(cosec\theta -cot\theta )}$

$\frac{cose{c}^2\theta -co{t}^2\theta }{cosec\theta -cot\theta }=\frac{1}{si{n}^2\theta }-\frac{co{s}^2\theta }{si{n}^2\theta }[\because cosec\theta =\frac{1}{sin\theta }cot\theta =\frac{cosec\theta }{sin\theta }]$

$\frac{\frac{1-co{s}^2\theta }{si{n}^2\theta }}{cosec\theta -cot\theta }=\frac{1}{cosec\theta -cot\theta }[\because 1-co{s}^2\theta =si{n}^2\theta ]$

$\because L.H.S =\frac{1}{cosec\theta-cot\theta} = RHS.