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8. Introduction to Trigonometry
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Prove that $\left(\sin ^{4} \theta-\cos ^{4} \theta+1\right) \operatorname{cosec}^{2} \theta=2$
Option A
Option B
Option C
Option D
Solution
L.H.S.$=\left(\sin ^{4} \theta-\cos ^{4} \theta+1\right) \operatorname{cosec}^{2} \theta$
$=\left[\left(\sin ^{2} \theta-\cos ^{2} \theta\right)\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+1\right] \operatorname{cosec}^{2} \theta$
$=\left(\sin ^{2} \theta-\cos ^{2} \theta+1\right) \operatorname{cosec}^{2} \theta$
$\quad\left[\right.$ Because $\left.\sin ^{2} \theta+\cos ^{2} \theta=1\right]$
$=2 \sin ^{2} \theta \operatorname{cosec}^{2} \theta \quad\left[\right.$ Because $\left.1-\cos ^{2} \theta=\sin ^{2} \theta\right]$
$=2=$ R.H.S.
Standard 10
Mathematics
