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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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