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
$2 \sqrt{3}$
$\frac{3}{2}$
$\sqrt{3}$
$\frac{1}{2}$
The equation $3{\sin ^2}x + 10\cos x - 6 = 0$ is satisfied, if
Statement $-1:$ The number of common solutions of the trigonometric equations $2\,sin^2\,\theta - cos\,2\theta = 0$ and $2 \,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, 2\pi ]$ is two.
Statement $-2:$ The number of solutions of the equation, $2\,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, \pi ]$ is two.
The solution of $\frac{1}{2} +cosx + cos2x + cos3x + cos4x = 0$ is
Let $A = \left\{ {\theta \,:\,\sin \,\left( \theta \right) = \tan \,\left( \theta \right)} \right\}$ and $B = \left\{ {\theta \,:\,\cos \,\left( \theta \right) = 1} \right\}$ be two sets. Then
If $2{\cos ^2}x + 3\sin x - 3 = 0,\,\,0 \le x \le {180^o}$, then $x =$