If $\cos \left( {{{\tan }^{ - 1}}x + {{\cot }^{ - 1}}\sqrt 3 } \right) = 0$, then the value of $x$ is

We have, $\cos \left( {{{\tan }^{ - 1}}x + {{\cot }^{ - 1}}\sqrt 3 } \right) = 0$

$\Rightarrow$ ${\tan ^{ - 1}}x + {\cot ^{ - 1}}\sqrt 3 = {\cos ^{ - 1}}0$

$\Rightarrow$ ${\tan ^{ - 1}}x + {\cot ^{ - 1}}\sqrt 3 = {\cos ^{ - 1}}\cos \frac{\pi }{2}$

$\Rightarrow$ ${\tan ^{ - 1}}x + {\cot ^{ - 1}}\sqrt 3 = \frac{\pi }{2}$

$\Rightarrow$ ${\tan ^{ - 1}}x = \frac{\pi }{2} - {\cot ^{ - 1}}\sqrt 3$
$\Rightarrow$ ${\tan ^{ - 1}}x = {\tan ^{ - 1}}\sqrt 3$

$therefore,$ $x = \sqrt 3$