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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:

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