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

$\begin{array}{1 1} \sqrt 3 \\ - \sqrt 3 \\ -\frac{1}{3} \\ \frac{1}{\sqrt 3 } \end{array}$

Toolbox:
• $tan^{-1}x+cot^{-1}x=\large\frac{\pi}{2}$
• $cos \large\frac{\pi}{2} = 0\:\:\:or\:cos^{-1}0=\large\frac{\pi}{2}$
Ans:$\sqrt 3$

The given equation can be written as
$tan^{-1}x+cot^{-1}\sqrt3=cos^{-1}0$

But $cos^{-1}0=\large\frac{\pi}{2}$
$\Rightarrow\:tan^{-1}x+cot^{-1}\sqrt3=\large\frac{\pi}{2}$

From the above formula we can say that
$\Rightarrow\: x = \sqrt 3$

edited Mar 16, 2013