Prove the following trigonometric identities:$ \frac{\left(1+\tan ^{2} \theta\right) \cot \theta}{\operatorname{cosec}^{2} \theta}=\tan \theta $

To do:

We have to prove that \( \frac{\left(1+\tan ^{2} \theta\right) \cot \theta}{\operatorname{cosec}^{2} \theta}=\tan \theta \).

Solution:

We know that,

$\sec ^{2} \theta-\tan^2 \theta=1$.......(i)

$\cot \theta=\frac{\operatorname{cosec} \theta}{\sec \theta}$.......(ii)

$\tan \theta=\frac{\sec \theta}{\operatorname{cosec} \theta}$.........(iii)

Therefore,

$\frac{\left(1+\tan ^{2} \theta\right) \cot \theta}{\operatorname{cosec}^{2} \theta}=\frac{\left(\sec ^{2} \theta\right) \frac{\operatorname{cosec} \theta}{\sec \theta}}{\operatorname{cosec}^{2} \theta}$                                [From (i) and (ii)]

$=\frac{\sec \theta}{\operatorname{cosec} \theta}$                

$=\tan \theta$                  [From (iii)]

Hence proved.    

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Updated on: 10-Oct-2022

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