The number of solutions of the given equation $\tan \theta + \sec \theta = \sqrt 3 ,$ where $0 < \theta < 2\pi $ is

Number of solutions to the system of equations $sin \frac{x+y}{2}=0$ and $|x| + |y| = 1$

Let $f:[0,2] \rightarrow R$ be the function defined by

$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$

If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . . 

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

Number of solutions of $8cosx$ = $x$ will be 

Prove that

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