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  4. If cot-1 x+sin-1 ((1/√5) ) = (π/4), then x is equal to

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Q. If $cot^{-1} x+sin^{-1} \left(\frac{1}{\sqrt{5}} \right) = \frac{\pi}{4}$, then $x$ is equal to

Inverse Trigonometric Functions

Solution:

$\because cot^{-1} x+sin^{-1} \frac{1}{\sqrt{5}} = \frac{\pi}{4}$
$\Rightarrow tan^{-1} \frac{1}{x} + tan^{-1} \frac{1}{2} = tan^{-1}\,1$
$\Rightarrow tan^{-1} \frac{1}{x} = tan^{-1}\,1 - tan^{-1} \frac{1}{2} = tan^{-1}\left(\frac{1-\frac{1}{2}}{1+\frac{1}{2}}\right)$
$\Rightarrow tan^{-1} \frac{1}{x} = tan^{-1} \frac{1}{3}$
$\Rightarrow x = 3$