Number of roots of the equation ${\cos ^2}x + \frac{{\sqrt 3  + 1}}{2}\sin x - \frac{{\sqrt 3 }}{4} - 1 = 0$ which lie in the interval $[-\pi,\pi ]$ is

If $(1 + \tan \theta )(1 + \tan \phi ) = 2$, then $\theta + \phi  =$ ....$^o$

The number of values of $x$ for which $sin2x + sin4x = 2$ is

The numbers of solution $(s)$ of the equation $\left( {1 - \frac{1}{{2\,\sin x}}} \right){\cos ^2}\,2x\, = \,2\,\sin x\, - \,3\, + \,\frac{1}{{\sin x}}$ in $[0,4\pi ]$ is

Let $A=\left\{\theta \in R:\left(\frac{1}{3} \sin \theta+\frac{2}{3} \cos \theta\right)^2=\frac{1}{3} \sin ^2 \theta+\frac{2}{3} \cos ^2 \theta\right\}$.Then

Prove that

$\cos 2 x \cos \frac{x}{2}-\cos 3 x \cos \frac{9 x}{2}=\sin 5 x \sin \frac{5 x}{2}$