Number of roots of the equation $\cos^2x+\dfrac {\sqrt 3+1}{2} \sin x-\dfrac {\sqrt 3}{4}-1=0$ which lie in the interval $[-\pi, \pi]$ is-

Replace cos^2(x) with 1 - sin^2(x) to form the quadratic equation 4sin^2(x) - (sqrt(3)+1)2sin(x) + sqrt(3) = 0. Factoring this gives (sin(x) - 1)(4sin(x) - sqrt(3)) = 0, so sin(x) = 1 or sin(x) = sqrt(3)/2. In the interval [-pi, pi], sin(x) = 1 gives the single solution pi, while sin(x) = sqrt(3)/2 gives the three solutions -2pi/3, pi/3, and 2pi/3, making a total of 4 roots.

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