Number of solutions of the equation cos theta | vedclass

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(D) Given equation: $\cos 	heta + \cos 2	heta - \sqrt{3}(\sin 	heta + \sin 2	heta) + 1 = 0$. Using $\cos 2	heta = 2\cos^2 	heta - 1$ and $\sin 2

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$4$

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(D) Given equation: $\cos 	heta + \cos 2	heta - \sqrt{3}(\sin 	heta + \sin 2	heta) + 1 = 0$. Using $\cos 2	heta = 2\cos^2 	heta - 1$ and $\sin 2	heta = 2\sin 	heta \cos 	heta$: $\cos 	heta + 2\cos^2 	heta - 1 - \sqrt{3}\sin 	heta - 2\sqrt{3}\sin 	heta \cos 	heta + 1 = 0$. $\cos 	heta(1 + 2\cos 	heta) - \sqrt{3}\sin 	heta(1 + 2\cos 	heta) = 0$. $(1 + 2\cos 	heta)(\cos 	heta - \sqrt{3}\sin 	heta) = 0$. Case $1$: $1 + 2\cos 	heta = 0 \implies \cos 	heta = -1/2$. In $(0, 2\pi)$,$	heta = 2\pi/3, 4\pi/3$. Case $2$: $\cos 	heta - \sqrt{3}\sin 	heta = 0 \implies 	an 	heta = 1/\sqrt{3}$. In $(0, 2\pi)$,$	heta = \pi/6, 7\pi/6$. The solutions are $\pi/6, 2\pi/3, 7\pi/6, 4\pi/3$. Total number of solutions is $4$.

Number of solutions of the equation $\cos \theta + \cos 2\theta - \sqrt{3}(\sin \theta + \sin 2\theta) + 1 = 0$ lying in the interval $(0, 2\pi)$ is

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