The number of solutions of the equation $1 + {\sin ^4}\,x = {\cos ^2}\,3x,x\,\in \,\left[ { - \frac{{5\pi }}{2},\frac{{5\pi }}{2}} \right]$ is
$3$
$4$
$5$
$7$
Number of principal solution of the equation $tan \,3x - tan \,2x - tan\, x = 0$, is
The set of angles btween $0$ & $2\pi $ satisfying the equation $4\, cos^2 \, \theta - 2 \sqrt 2 \, cos \,\theta - 1 = 0$ is
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 $.............$.
The general solution of $\sin x - \cos x = \sqrt 2 $, for any integer $n$ 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