The number of solutions of the equation $2 \theta-\cos ^{2} \theta+\sqrt{2}=0$ is $R$ is equal to

If $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$, then $\sin \left( {\theta + \frac{\pi }{4}} \right)$ equals

If $\cos 7\theta = \cos \theta - \sin 4\theta $, then the general value of $\theta $ is

The set of values of $‘a’$ for which the equation, $cos\, 2x + a\, sin\, x = 2a - 7$ possess a solution is :

The number of elements in the set $S =\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^2 \theta+2=0\right\}$ is $...........$.

If $0 \le x \le \pi $ and ${81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30$, then $x =$