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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Updated on: 10-Oct-2022

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