If ${\sin ^2}\theta - 2\cos \theta + \frac{1}{4} = 0,$ then the general value of $\theta $ is

  • A

    $n\pi \pm \frac{\pi }{3}$

  • B

    $2n\pi \pm \frac{\pi }{3}$

  • C

    $2n\pi \pm \frac{\pi }{6}$

  • D

    $n\pi \pm \frac{\pi }{6}$

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Find the general solution of the equation $\cos 3 x+\cos x-\cos 2 x=0$

Statement $-1:$ The number of common solutions of the trigonometric equations $2\,sin^2\,\theta - cos\,2\theta  = 0$ and $2 \,cos^2\,\theta - 3\,sin\,\theta  = 0$ in the interval $[0, 2\pi ]$ is two.

Statement $-2:$ The number of solutions of the equation, $2\,cos^2\,\theta  - 3\,sin\,\theta  = 0$ in the interval $[0, \pi ]$ is two.

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Let $\theta, 0 < \theta < \pi / 2$, be an angle such that the equation $x ^2+4 x \cos \theta+\cot \theta=0$ has equal roots for $x$. Then $\theta$ in radians is

  • [KVPY 2021]