$\sqrt 3 ({\cos ^2}x) = (\sqrt 3 - 1)\cos x + 1$ $\Rightarrow$ $\sqrt 3 {\cos ^2}x - \sqrt 3 \cos x + \cos x - 1 = 0$ $\Rightarrow \sqrt 3 \cos x(\cos x - 1) + (\cos x - 1) = 0$ $\Rightarrow (\cos x - 1)(\sqrt 3 \cos x + 1) = 0$ $\cos x = 1$ $\Rightarrow x = 0$ $[as x \in \left[ {0,{\pi \over 2}} \right]$] and $\cos x = - {1 \over {\sqrt 3 }}$ (not possible in $x \in \left[ {0,{\pi \over 2}} \right]$] $\therefore$ Number of solution = 1