The number of elements in the set $S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$ is$.....$

The general value $\theta $ is obtained from the equation $\cos 2\theta = \sin \alpha ,$ is

The general solution of $sin\, x + sin \,5x = sin\, 2x + sin \,4x$ is :

The number of solutions of the equation $4 \sin ^2 x-4$ $\cos ^3 \mathrm{x}+9-4 \cos \mathrm{x}=0 ; \mathrm{x} \in[-2 \pi, 2 \pi]$ is :

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

If ${\sin ^2}\theta - 2\cos \theta + \frac{1}{4} = 0,$ then the general value of $\theta $ is