Question:
Prove the following trigonometric identities.
$\operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=1$
Solution:
We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$
So,
$\operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=\operatorname{cosec} \theta \sqrt{\sin ^{2} \theta}$
$=\operatorname{cosec} \theta \sin \theta$
$=\frac{1}{\sin \theta} \times \sin \theta$
$=1$