If sqrt{3}(cos ^{2} x)=(sqrt{3}-1) cos x+1, t | vedclass

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(A) Given equation: $\sqrt{3} \cos^2 x = (\sqrt{3}-1) \cos x + 1$Rearranging the terms: $\sqrt{3} \cos^2 x - \sqrt{3} \cos x + \cos x - 1 = 0$Factoriz

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(A) Given equation: $\sqrt{3} \cos^2 x = (\sqrt{3}-1) \cos x + 1$Rearranging the terms: $\sqrt{3} \cos^2 x - \sqrt{3} \cos x + \cos x - 1 = 0$Factorizing by grouping: $\sqrt{3} \cos x (\cos x - 1) + 1 (\cos x - 1) = 0$$(\sqrt{3} \cos x + 1)(\cos x - 1) = 0$This gives two cases:Case $1$: $\cos x = 1 \Rightarrow x = 0$ (which is in the interval $[0, \frac{\pi}{2}]$)Case $2$: $\cos x = -\frac{1}{\sqrt{3}}$. Since $\cos x$ is negative in the second quadrant and $x \in [0, \frac{\pi}{2}]$,there is no solution for this case in the given interval.Thus,there is only $1$ solution,which is $x = 0$.

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

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