Question

1/cosec_cot=cosec+cot​

Answer 1

hey mate here is ur answer

see the attachments

hope this helps

if u have any doubts ask in comments

Answer 2

We know that

$\frac{1}{cosec \: \theta - cot \: \theta}$
can be solved by rationalising the denominator

$\frac{1}{cosec \: \theta - cot \: \theta} \times \frac{cosec \: \theta + cot \: \theta}{cosec \: \theta + cot \: \theta} \\ \\ \\ = > \frac{cosec \: \theta + cot \: \theta}{ {cosec}^{2} \theta - {cot}^{2} \theta }$
As we also know that

${cosec}^{2} \theta - {cot}^{2} \theta = 1$
So
$= > \frac{cosec \: \theta + cot \: \theta}{ {cosec}^{2} \theta - {cot}^{2} \theta } \\ \\ \\ = > \frac{cosec \: \theta + cot \: \theta}{1} \\ \\ \\ = > cosec \: \theta + cot \: \theta$

= RHS

hence proved