Question

The value of expression : $\tan ^{-1}\left(\frac{\sqrt{2}}{2}\right)+\sin ^{-1}\left(\frac{\sqrt{5}}{5}\right)-\cos ^{-1}\left(\frac{\sqrt{10}}{10}\right)$ is

• A. $\cot ^{-1}\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)$
• B. $\cot ^{-1}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)$
• C. $-\pi+\cot ^{-1}\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)$
• D. $\pi-\cot ^{-1}\left(\frac{1-\sqrt{2}}{1+\sqrt{2}}\right)$

Answer 1

Answer: (C)

$\tan ^{-1} \frac{1}{\sqrt{2}}+\sin ^{-1} \frac{1}{\sqrt{5}}-\cos ^{-1} \frac{1}{\sqrt{10}} \Rightarrow \tan ^{-1} \frac{1}{\sqrt{2}}+\tan ^{-1} \frac{1}{2}-\tan ^{-1} 3$
$\tan ^{-1} \frac{1}{\sqrt{2}}-\left[\tan ^{-1} 3-\tan ^{-1} \frac{1}{2}\right] \Rightarrow \tan ^{-1} \frac{1}{\sqrt{2}}-\left[\tan ^{-1} \frac{3-\frac{1}{2}}{1+\frac{3}{2}}\right]=\left(\tan ^{-1} \frac{1}{\sqrt{2}}\right)\left(\tan ^{-1} 1\right) $
$=-\left[\tan ^{-1} 1-\tan ^{-1} \frac{1}{\sqrt{2}}\right]=-\left[\tan ^{-1} \frac{1-\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}\right]=-\tan ^{-1} \frac{\sqrt{2}-1}{\sqrt{2}+1} $
$=-\cot ^{-1} \frac{\sqrt{2}+1}{\sqrt{2}-1}=-\left[\cot ^{-1}-\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)\right]=-\left[\pi-\cot ^{-1} \frac{1+\sqrt{2}}{1-\sqrt{2}}\right]=-\pi+\cot ^{-1} \frac{1+\sqrt{2}}{1-\sqrt{2}} \Rightarrow \text { (C) }$