Question
Prove the following trigonometric identities.
$(\text{cosec}\theta+\sin\theta)(\text{cosec}\theta-\sin\theta)=\cot^2\theta+\cos^2\theta$
$(\text{cosec}\theta+\sin\theta)(\text{cosec}\theta-\sin\theta)=\cot^2\theta+\cos^2\theta$
✓
Answer
$\text{L.H.S}=(\text{cosec}\theta+\sin\theta)(\text{cosec}\theta-\sin\theta)$
$=(\text{cosec}^2\theta-\sin^2\theta)$
$=(1+\cot^2\theta)-(1-\cos^2\theta)$
$=1+\cot^2\theta-1+\cos^2\theta$
$=\cot^2\theta+\cos^2\theta=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
$=(\text{cosec}^2\theta-\sin^2\theta)$
$=(1+\cot^2\theta)-(1-\cos^2\theta)$
$=1+\cot^2\theta-1+\cos^2\theta$
$=\cot^2\theta+\cos^2\theta=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
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