The number of solutions of the equation $\sin \theta+\cos \theta=\sin 2 \theta$ in the interval $[-\pi, \pi]$ is
$1$
$2$
$3$
$4$
The number of solutions of the pair of equations $ 2 \sin ^2 \theta-\cos 2 \theta=0 $, $ 2 \cos ^2 \theta-3 \sin \theta=0$ in the interval $[0,2 \pi]$ is
The solution of $tan\,\, 2\theta\,\, tan\theta = 1$ is
The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is
The solution of $\frac{1}{2} +cosx + cos2x + cos3x + cos4x = 0$ is
Let $\theta \in [0, 4\pi ]$ satisfy the equation $(sin\, \theta + 2) (sin\, \theta + 3) (sin\, \theta + 4) = 6$ . If the sum of all the values of $\theta $ is of the form $k\pi $, then the value of $k$ is