Prove the following
Question:

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

Solution:

$\sqrt{3} t^{2}-(\sqrt{3}-1) t-1=0$ (where $t=\cos x$ )

Now, $t=\frac{(\sqrt{3}-1) \pm \sqrt{4+2 \sqrt{3}}}{2 \sqrt{3}}$

$t=\cos x=1$ or $-\frac{1}{\sqrt{3}} \rightarrow$ rejected as $x \in\left[0, \frac{\pi}{2}\right]$

$\Rightarrow \cos x=1$

$\Rightarrow$ No. of solution $=1$