Question:
Prove the following trigonometric identities.
$\left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=1$
Solution:
We know that, $\sin ^{2} A+\cos ^{2} A=1$
so
$\left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=\sin ^{2} A \operatorname{cosec}^{2} A$
$=(\sin A \operatorname{cosec} A)^{2}$
$=\left(\sin A \times \frac{1}{\sin A}\right)^{2}$
$=(1)^{2}$
$=1$