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

If $e ^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \ldots \infty\right) \log _{e} 2}$ satisfies the equation $t ^{2}-9 t +8=0,$ then the value of $\frac{2 \sin x}{\sin x+\sqrt{3} \cos x}\left(0 < x < \frac{\pi}{2}\right)$ 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

The general solution of the equation $(\sqrt 3 - 1)\sin \theta + (\sqrt 3 + 1)\cos \theta = 2$ is

If $\tan 2\theta \tan \theta = 1$, then the general value of $\theta $ is

If $x = \frac{{n\pi }}{2}$ , satisfies the equation $sin\, \frac{x}{2}- cos \frac{x}{2} = 1$ $- sin\, x$ & the inequality $\left| {\frac{x}{2}\,\, - \,\,\frac{\pi }{2}} \right|\,\, \le \,\,\frac{{3\pi }}{4}$, then: