2 - ( ^ -1 4 5 + ^ -1 5 13 + ^ -1 16 65 ) is equal to - Vidyadip

MCQ

$2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to

A
$\frac{7 \pi}{4}$
B
$\frac{5 \pi}{4}$
✓
$\frac{3 \pi}{2}$
D
$\frac{\pi}{2}$

✓

Answer

Correct option: C.

$\frac{3 \pi}{2}$

c
$2 \pi-\left(\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{16}{65}\right)\right)$

$=2 \pi-\left(\tan ^{-1}\left(\frac{4}{3}\right)+\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{16}{63}\right)\right)$

$=2 \pi-\left(\tan ^{-1}\left(\frac{63}{16}\right)+\tan ^{-1}\left(\frac{16}{63}\right)\right)$

$=2 \pi-\frac{\pi}{2}=\frac{3 \pi}{2}$

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