Q53 / 120 · VITEEE 2009 · Mathematics
$\cos ^{-1}\left(\frac{-1}{2}\right)-2 \sin ^{-1}\left(\frac{1}{2}\right)+3 \cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$
$-4 \tan ^{-1}(-1)$ equals
- A$\frac{19 \pi}{12}$
- B$\frac{35 \pi}{12}$
- C$\frac{47 \pi}{12}$
- Correct answerD$\frac{43 \pi}{12}$
Solution
$$
\begin{array}{l}
\cos ^{-1}\left(-\frac{1}{2}\right)-2 \sin ^{-1}\left(\frac{1}{2}\right)+3 \cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right) \\
=\pi-\cos ^{-1}\left(\frac{1}{2}\right)-2\left(\frac{\pi}{6}\right)+3\left(\pi-\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)\right) \\
=\frac{\pi}{3}+3 \cdot \frac{3 \pi}{4}+\pi=\frac{43 \pi}{12}
\end{array}
$$
Hence, the correct answer is option: 'D'.