Question:
Write the value of $\operatorname{cosec}^{2}\left(90^{\circ}-\theta\right)-\tan ^{2} \theta$.
Solution:
We have,
$\operatorname{cosec}^{2}\left(90^{\circ}-\theta\right)-\tan ^{2} \theta=\left\{\operatorname{cosec}\left(90^{\circ}-\theta\right)\right\}^{2}-\tan ^{2} \theta$
$=(\sec \theta)^{2}-\tan ^{2} \theta$
$=\sec ^{2} \theta-\tan ^{2} \theta$
We know that, $\sec ^{2} \theta-\tan ^{2} \theta=1$
Therefore, $\operatorname{cosec}^{2}\left(90^{\circ}-\theta\right)-\tan ^{2} \theta=1$
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