Prove the following trigonometric identities:$ (\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\cot ^{2} \theta+\cos ^{2} \theta $

To do:

We have to prove that \( (\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\cot ^{2} \theta+\cos ^{2} \theta \).

Solution:

We know that,

$\operatorname{cosec}^2 \theta-\cot^2 \theta=1$........(i)

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

Therefore,

$(\operatorname{cosec} \theta+\sin \theta)(\operatorname{cosec} \theta-\sin \theta)=\operatorname{cosec} ^{2} \theta-\sin^2 \theta$       [$(a+b)(a-b)=a^2-b^2$]

$=(1+\cot^2 \theta)-(1-\cos^2 \theta)$     (From (i) and (ii))

$=1-1+\cot^2 \theta+\cos^2 \theta$    

$=\cot^2 \theta+\cos^2 \theta$       

Hence proved.   

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Updated on: 10-Oct-2022

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