Question

Solve for $x:{\left\{x\cos \left({\cot }^{-1}x\right)+\sin \left({\cot }^{-1}x\right)\right\}}^2=\frac{51}{50}$,

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{5\sqrt{2}}$
• C. $2\sqrt{2}$
• D. $5\sqrt{2}$

Answer 1

Answer: (B)

${\left\{x\cos \left({\cot }^{-1}x\right)+\sin \left({\cot }^{1}x\right)\right\}}^2=\frac{51}{50}$
$\Rightarrow {\left\{x\cos \left({\tan }^{-1}\frac{1}{x}\right)+\sin \left({\tan }^{-1}\frac{1}{x}\right)\right\}}^2=\frac{51}{50}$
$\Rightarrow \left\{x\cos \left[{\cos }^{-1}\left(\frac{1}{\sqrt{1+\frac{1}{x^2}}}\right)\right]+\sin \left({\sin }^{-1}\left(\frac{\frac{1}{x}}{\sqrt{1+\frac{1}{x^2}}}\right)\right)\right\}=\frac{51}{50}$
$\Rightarrow {\left(\frac{x^2}{\sqrt{1+x^2}}+\frac{1}{\sqrt{1+x^2}}\right)}^2=\frac{51}{50}$
$\Rightarrow \frac{{\left(x^2+1\right)}^2}{\left(x^2+1\right)}=\frac{51}{50}$
$\Rightarrow x^2+1=\frac{51}{50}$
$\Rightarrow x=\pm \frac{1}{5\sqrt{2}}$