Question

The value of $2cos^{- 1} \sqrt{\frac{2}{3}}-2cos^{- 1} ⁡ \frac{\sqrt{6} + 1}{2 \sqrt{3}}$ is equal to

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (A)

$2 \cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)-2 \cos ^{-1}\left(\frac{\sqrt{6}+1}{2 \sqrt{3}}\right)$
$=2 \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{3}-\sqrt{2}}{1+\sqrt{6}}\right)$
$=2 \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-2\left[\tan ^{-1} \sqrt{3}-\tan ^{-1} \sqrt{2}\right]$
$=2\left(\tan ^{-1} \frac{1}{\sqrt{2}}+\tan ^{-1} \sqrt{2}\right)-2 \tan ^{-1} \sqrt{3}$
$=\pi-\frac{2 \pi}{3}=\frac{\pi}{3}$