The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations

$ (y+z) \cos 3 \theta=(x y z) \sin 3 \theta $

$ x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z} $

$ (x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$ have a solution $\left(\mathrm{x}_0, \mathrm{y}_0, \mathrm{z}_0\right)$ with $\mathrm{y}_0 \mathrm{z}_0 \neq 0$, is

If $\tan \theta + \tan 2\theta + \tan 3\theta = \tan \theta \tan 2\theta \tan 3\theta $, then the general value of $\theta $ is

The general value of $\theta $  that satisfies both the equations $cot^3\theta + 3 \sqrt 3 $ = $0$ & $cosec^5\theta + 32$ = $0$ is $(n \in  I)$

If $sin^4\,\,\alpha + 4\,cos^4\,\,\beta + 2 = 4\sqrt 2\,\,sin\,\alpha \,cos\,\beta ;$ $\alpha \,,\,\beta \, \in \,[0,\pi ],$ then $cos( \alpha + \beta)$ is equal to

Values of $\theta (0 < \theta < {360^o})$ satisfying ${\rm{cosec}}\theta + 2 = 0$ are

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  = $