The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta$ is

If $\cos \theta + \cos 7\theta + \cos 3\theta + \cos 5\theta = 0$, then $\theta $

The sum of solutions in $x \in (0,2\pi )$ of the equation, $4\cos (x).\cos \left( {\frac{\pi }{3} - x} \right).\cos \left( {\frac{\pi }{3} + x} \right) = 1$ is equal to 

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2 \theta \cos \frac{\theta}{2}=\cos 3 \theta \cos \frac{9 \theta}{2}$, then $mn$ is equal to $.............$.

If $\cos 2\theta + 3\cos \theta = 0$, then the general value of $\theta $ is

If $\alpha ,\beta ,\gamma $ be the angles made by a line with $x, y$ and $z$ axes respectively so that $2\left( {\frac{{{{\tan }^2}\,\alpha }}{{1 + {{\tan }^2}\,\alpha }} + \frac{{{{\tan }^2}\,\beta }}{{1 + {{\tan }^2}\,\beta }} + \frac{{{{\tan }^2}\,\gamma }}{{1 + {{\tan }^2}\,\gamma }}} \right) = 3\,{\sec ^2}\,\frac{\theta }{2},$ then $\theta =$