Prove the following trigonometric identities:$ \left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=1 $

To do:

We have to prove that \( \left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=1 \).

Solution:
We know that,

$ \sin ^{2} A+\cos ^{2} A=1$.......(i)

$\sin ^{2} A\times\operatorname{cosec}^{2} A=1$.......(ii)

Therefore,

$\left(1-\cos ^{2} A\right) \operatorname{cosec}^{2} A=(\sin ^{2} A)(\operatorname{cosec}^{2} A)$       (From (i))

$=\sin ^{2} A\times\operatorname{cosec}^{2} A$              (From (ii))

$=1$

Hence proved.

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Updated on: 10-Oct-2022

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