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Prove the following trigonometric identities:
\( \operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=1 \)
To do:
We have to prove that \( \operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=1 \).
Solution:
We know that,
$\sin ^{2} A+cos ^{2} A=1$.......(i)
$ \sin\ A\times\operatorname{cosec} A=1$.......(ii)
Therefore,
$\operatorname{cosec} \theta \sqrt{1-\cos ^{2} \theta}=\operatorname{cosec} \theta \sqrt{\sin ^{2} \theta}$ (From (i))
$=\operatorname{cosec} \theta \sin \theta$
$=1$ (From (ii))
Hence proved.
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