The number of real solutions $x$ of the equation $\cos ^2(x \sin (2 x))+\frac{1}{1+x^2}=\cos ^2 x+\sec ^2 x$ is

If $\alpha ,\,\beta ,\,\gamma $ and $\delta $ are the solutions of the equation $\tan \left( {\theta  + \frac{\pi }{4}} \right) = 3\,\tan \,3\theta $ , no two of which have equal tangents, then the value of $tan\, \alpha  + tan\, \beta + tan\, \gamma + tan\, \delta $ is

If $\sin \theta + \cos \theta = \sqrt 2 \cos \alpha $, then the general value of $\theta $ is

The solution of the equation $\left| {\,\begin{array}{*{20}{c}}{\cos \theta }&{\sin \theta }&{\cos \theta }\\{ - \sin \theta }&{\cos \theta }&{\sin \theta }\\{ - \cos \theta }&{ - \sin \theta }&{\cos \theta }\end{array}\,} \right| = 0$, is

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

The general value of $\theta $satisfying the equation $2{\sin ^2}\theta - 3\sin \theta - 2 = 0$ is