Question

If $\sqrt{3}\left({\cos }^{2}x\right)=(\sqrt{3}-1)\cos x+1,$ the number of solutions of the given equation when $x\in \left[0,\frac{\pi }{2}\right]$ is

Answer 1

Answer: 1

$\sqrt{3}(\cos x{)}^2-\sqrt{3}\cos x+\cos x-1=0$
$\Rightarrow (\sqrt{3}\cos x+1)(\cos x-1)=0$
$\Rightarrow \cos x=1$ or $\cos x=-\frac{1}{\sqrt{3}}$ (reject)
$\Rightarrow x=0$ only