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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.