If $\cos 2\theta = (\sqrt 2 + 1)\,\,\left( {\cos \theta – \frac{1}{{\sqrt 2 }}} \right)$, then the value of $\theta $ is

The general solution of $\sin x – \cos x = \sqrt 2 $, for any integer $n$ is

$\tan \,{20^o}\cot \,{10^o}\cot \,{50^o}$ is equal to

If $\sin 2\theta = \cos \theta ,\,\,0 < \theta < \pi $, then the possible values of $\theta $ are

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. . . . . . . . .