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Write the value of cosec2 (90° − θ) − tan2 θ.

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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