Question

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

• A. $\frac{19\pi}{12}$
• B. $\frac{35\pi}{12}$
• C. $\frac{47\pi}{12}$
• D. $\frac{43\pi}{12}$

Hint:

$\cos^{-1}(-x) = \pi - \cos^{-1}x$.

Answer 1

The Correct Option is D

Step 1: Principal values
$\cos^{-1}(-1/2) = \pi - \pi/3 = 2\pi/3$. $\sin^{-1}(1/2) = \pi/6$. $\cos^{-1}(-1/\sqrt{2}) = \pi - \pi/4 = 3\pi/4$. $\tan^{-1}(-1) = -\pi/4$.
Step 2: Substitution
$(2\pi/3) - 2(\pi/6) + 3(3\pi/4) - 4(-\pi/4)$.
Step 3: Calculation
$= 2\pi/3 - \pi/3 + 9\pi/4 + \pi = \pi/3 + 9\pi/4 + \pi = \frac{4\pi + 27\pi + 12\pi}{12} = \frac{43\pi}{12}$.
Final Answer: (D)