Answer
$x = \frac{\sqrt{3}}{3}$
Work Step by Step
$arccot~x+2~arcsin~\frac{\sqrt{3}}{2} = \pi$
$arccot~x+(2)(\frac{\pi}{3}) = \pi$
$arccot~x = \frac{\pi}{3}$
$x = cot(\frac{\pi}{3})$
$x = \frac{1}{\sqrt{3}}$
$x = \frac{1}{\sqrt{3}}~\times~\frac{\sqrt{3}}{\sqrt{3}}$
$x = \frac{\sqrt{3}}{3}$
