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Solve the Following Questions

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

Solution:

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

$\Rightarrow(\sqrt{3} \cos x+1)(\cos x-1)=0$

$\Rightarrow \cos x=1$ or $\cos x=-\frac{1}{\sqrt{3}}$ (reject)

$\Rightarrow x=0$ only