Prove the following trigonometric identities.

We know that,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

So,

$\left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=\tan ^{2} \theta \times \cot ^{2} \theta$

$=(\tan \theta \times \cot \theta)^{2}$

$=\left(\tan \theta \times \frac{1}{\tan \theta}\right)^{2}$

$=(1)^{2}$

$=1$