Write 'True' or 'False' and justify your answer in each of the following:
$ \sqrt{\left(1-\cos ^{2} \theta\right) \sec ^{2} \theta}=\tan \theta $

Given:

\( \sqrt{\left(1-\cos ^{2} \theta\right) \sec ^{2} \theta}=\tan \theta \)

To do:

We have to find whether the given statement is true or false.

Solution:

We know that,

$\sin ^{2} \theta+\cos ^{2} \theta=1$

$\operatorname{sec}^{2} \theta = \frac{1}{\cos ^{2} \theta}$

$\sec \theta =\frac{1}{\cos \theta}$

$\tan \theta=\frac{\sin \theta}{\cos \theta}$

Therefore,

$\sqrt{(1-\cos ^{2} \theta) \sec ^{2} \theta} =\sqrt{\sin ^{2} \theta . \sec ^{2} \theta}$

$=\sqrt{\sin ^{2} \theta . \frac{1}{\cos ^{2} \theta}}$

$=\sqrt{\tan ^{2} \theta}$

$=\tan \theta$

The given statement is true.

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Updated on: 10-Oct-2022

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