The equation $\sin x\cos x = 2$ has

If $\sin \theta  + 2\sin \phi  + 3\sin \psi  = 0$ and $\cos \theta  + 2\cos \phi  + 3\cos \psi  = 0$ , then the value of $\cos 3\theta  + 8\cos 3\phi  + 27\cos 3\psi  = $ 

Number of solution$(s)$ of the equation $ln(1 + sin^2x) = 1 -ln(5 + x^2)$ is -

Number of solutions of equation $secx = 1 + cosx + cos^2x + ........ \infty$ in $x \in [-50 \pi, 50 \pi]$ is -

The solution of the equation ${\cos ^2}x - 2\cos x = $ $4\sin x - \sin 2x,$ $\,(0 \le x \le \pi )$ is

Let $S=\{\theta \in[0,2 \pi): \tan (\pi \cos \theta)+\tan (\pi \sin \theta)=0\}$.

Then $\sum_{\theta \in S } \sin ^2\left(\theta+\frac{\pi}{4}\right)$ is equal to