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  4. cos-1 ((-1/2))-2 sin-1((1/2))+3 cos-1((-1/√2))-4 tan-1 (-1) equals

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Q. $cos^{-1} \left(\frac{-1}{2}\right)-2 sin^{-1}\left(\frac{1}{2}\right)+3 cos^{-1}\left(\frac{-1}{\sqrt{2}}\right)-4 tan^{-1 }\left(-1\right) $ equals

VITEEEVITEEE 2009Inverse Trigonometric Functions

Solution:

$\cos ^{-1}\left(-\frac{1}{2}\right)-2 \sin ^{-1}\left(\frac{1}{2}\right)+3 \cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)-4 \tan ^{-1}(-1)$
$=\pi-\cos ^{-1}\left(\frac{1}{2}\right)-2\left(\frac{\pi}{6}\right)+3\left(\pi-\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)+4 \tan ^{-1}(1)$
$=\pi-\frac{\pi}{3}-\frac{\pi}{3}+3\left(\pi-\frac{\pi}{4}\right)+4 . \frac{\pi}{4}$
$=\frac{\pi}{3}+3 \pi-\frac{3 \pi}{4}+\pi$
$=\frac{43 \pi}{12}$