Prove the following trigonometric identities.

Given:

$\operatorname{cosec} \theta+\cot \theta=m$

$\operatorname{cosec} \theta-\cot \theta=n$

We have to prove $m n=1$

We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$

Multiplying the two equations, we have

$(\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)=m n$

$\Rightarrow\left(\frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}\right)\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right)=m n$

$\Rightarrow \quad\left(\frac{1+\cos \theta}{\sin \theta}\right)\left(\frac{1-\cos \theta}{\sin \theta}\right)=m n$

$\Rightarrow \quad \frac{(1+\cos \theta)(1-\cos \theta)}{\sin ^{2} \theta}=m n$

$\Rightarrow \quad \frac{1-\cos ^{2} \theta}{\sin ^{2} \theta}=m n$

$\Rightarrow \quad \frac{\sin ^{2} \theta}{\sin ^{2} \theta}=m n$

$\Rightarrow \quad 1=m n$

$\Rightarrow \quad m n=1$

Hence proved.