Question

Solution of the equation ${\cot }^{-1}x+{\sin }^{-1}\frac{1}{\sqrt{5}}=\frac{\pi }{4}$ is:

• A. $x=3$
• B. $x=\frac{1}{\sqrt{5}}$
• C. $x=0$
• D. None

Answer 1

The correct option is A $x=3$

Given, ${\cot }^{-1}x+{\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right)=\frac{\pi }{4}$
$\Rightarrow {\cot }^{-1}x+{\cot }^{-1}\left(\frac{\sqrt{1-{\left(\frac{1}{\sqrt{5}}\right)}^2}}{\frac{1}{\sqrt{5}}}\right)=\frac{\pi }{4}$
$\Rightarrow {\cot }^{-1}x+{\cot }^{-1}2=\frac{\pi }{4}$
$\Rightarrow {\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{4}$
$\Rightarrow {\tan }^{-1}\left(\frac{\frac{1}{x}+\frac{1}{2}}{1-\frac{1}{x}\frac{1}{2}}\right)=\frac{\pi }{4}\Rightarrow \frac{2+x}{2x-1}=1\Rightarrow 2+x=2x-1$

$\Rightarrow x=3$