Prove the following trigonometric identities:$ \tan ^{2} \theta \cos ^{2} \theta=1-\cos ^{2} \theta $

To do:

We have to prove that \( \tan ^{2} \theta \cos ^{2} \theta=1-\cos ^{2} \theta \).

Solution: We know that,

$ \tan ^{2} A=\frac{\sin ^{2} A}{\cos ^{2} A}$.......(i)

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

Therefore,

$\tan ^{2} \theta \cos ^{2} \theta=\frac{\sin ^{2} \theta}{\cos ^{2} \theta}(\cos ^{2} \theta)$       (From (i))

$=\sin ^{2} \theta$

$=1-cos ^{2} \theta$              (From (ii))

Hence proved. 

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Updated on: 10-Oct-2022

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