Q. The value of $\sin ^{-1}\left[\cot \left(\sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sec ^{-1} \sqrt{2}\right)\right]=$

KCETKCET 2022 Report Error

A

0

B

$\frac{\pi}{4}$

C

$\frac{\pi}{6}$

D

$\frac{\pi}{2}$

Solution:

$\sin ^{-1}\left[\cot \left(\sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sec ^{-1} \sqrt{2}\right)\right]$
$=\sin ^{-1}\left[\cot \left(\sin ^{-1} \sqrt{\frac{4-2 \sqrt{3}}{8}}+\cos ^{-1} \frac{2 \sqrt{3}}{4}+\sec ^{-1} \sqrt{2}\right)\right]$
$=\sin ^{-1}\left[\cot \left(\sin ^{-1} \frac{\sqrt{3}-1}{2 \sqrt{2}}+\cos ^{-1} \frac{\sqrt{3}}{2}+\sec ^{-1} \sqrt{2}\right)\right]$
$=\sin ^{-1}\left[\cot \left(15^{0}+30^{0}+45^{0}\right)\right]=\sin ^{-1}\left(\cot 90^{0}\right)$
$=\sin ^{-1}(0)=0$

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