^2 ( ^ -1 1 2 )+ ^2 ( ^ -1 2 )+ ^2 (cosec^ -1 4 )= - Vidyadip

MCQ

$\sin ^2\left(\sin ^{-1} \frac{1}{2}\right)+\tan ^2\left(\sec ^{-1} 2\right)+\cot ^2\left(\operatorname{cosec}^{-1} 4\right)=$

✓
$\frac{73}{4}$
B
$\frac{37}{2}$
C
$\frac{89}{4}$
D
$19$

✓

Answer

Correct option: A.

$\frac{73}{4}$

(A) Let $\operatorname{cosec}^{-1} 4=\theta \Rightarrow \operatorname{cosec} \theta=4$
$\cot ^2 \theta=\operatorname{cosec}^2 \theta-1=(4)^2-1=15$
$\sin ^2\left(\sin ^{-1} \frac{1}{2}\right)+\tan ^2\left(\sec ^{-1} 2\right)+\cot ^2\left(\operatorname{cosec}^{-1} 4\right)$
$=\sin ^2 30^{\circ}+\tan ^2 60^{\circ}+\cot ^2 \theta$
$=\left(\frac{1}{2}\right)^2+(\sqrt{3})^2+15$
$=\frac{73}{4}$

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