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  4. Solve for x: x cos ( cot -1 x)+ sin ( cot -1 x) 2=(51/50),

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

Inverse Trigonometric Functions

Solution:

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