Question

The value of the 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

The correct option is C $-\pi +{\cot }^{-1}\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)$

${\tan }^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{5}}{5}\right)-{\cos }^{-1}\left(\frac{\sqrt{10}}{10}\right)={\tan }^{-1}\frac{1}{\sqrt{2}}+{\tan }^{-1}\frac{1}{2}-{\tan }^{-1}3={\tan }^{-1}\frac{1}{\sqrt{2}}-[{\tan }^{-1}3-{\tan }^{-1}\frac{1}{2}]={\tan }^{-1}\frac{1}{\sqrt{2}}-{\tan }^{-1}\left(\frac{3-\frac{1}{2}}{1+\frac{3}{2}}\right)={\tan }^{-1}\frac{1}{\sqrt{2}}-{\tan }^{-1}1={\tan }^{-1}\left(\frac{\frac{1}{\sqrt{2}}-1}{1+\frac{1}{\sqrt{2}}}\right)={\tan }^{-1}\left(\frac{1-\sqrt{2}}{1+\sqrt{2}}\right)$
As ${\tan }^{-1}x=-\pi +{\cot }^{-1}\frac{1}{x}$ when $x<0,$
${\tan }^{-1}\left(\frac{1-\sqrt{2}}{1+\sqrt{2}}\right)=-\pi +{\cot }^{-1}\frac{1+\sqrt{2}}{1-\sqrt{2}}$