If $\sin \theta + 2\sin \phi + 3\sin \psi = 0$ and $\cos \theta + 2\cos \phi + 3\cos \psi = 0$ , then the value of $\cos 3\theta + 8\cos 3\phi + 27\cos 3\psi = $
$\cos (3\theta + 3\phi + 3\psi )$
$18\cos (\theta + \phi + \psi )$
$6\cos (\theta + \phi + \psi )$
$36\cos (\theta + \phi + \psi )$
The solution of the equation $\sec \theta - {\rm{cosec}}\theta = \frac{4}{3}$ is
Let $S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2\right.$ $\left.\cos ^{2} 2 \theta=2\right\}$. Then, the sum of roots of all the equations $x ^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x +6 \sin ^{2} \theta=0$ $\theta \in S$, is$...$
The number of solutions to the equation $\cos ^4 x+\frac{1}{\cos ^2 x}=\sin ^4 x+\frac{1}{\sin ^2 x}$ in the interval $[0,2 \pi]$ is
If $|cos\ x + sin\ x| + |cos\ x\ -\ sin\ x| = 2\ sin\ x$ ; $x \in [0,2 \pi ]$ , then maximum integral value of $x$ is
The smallest positive root of the equation $tanx\, -\, x = 0$ lies on