The equation, $sin^2 \theta - \frac{4}{{{{\sin }^3}\,\,\theta \,\, - \,\,1}} = 1$$ -\frac{4}{{{{\sin }^3}\,\,\theta \,\, - \,\,1}}$ has :

If $\cos \,x = \frac{{2\cos y - 1}}{{2 - \cos y}},x,\,y\, \in \,\left( {0,\pi } \right),$ then $tan(x/2)cot(y/2) =$

The number of solutions to the equation $\cos ^4 x+\frac{1}{\cos ^2 x}=\sin ^4 x+\frac{1}{\sin ^2 x}$ in the interval $[0,2 \pi]$ is

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 $\alpha ,\beta ,\gamma $ be the angles made by a line with $x, y$ and $z$ axes respectively so that $2\left( {\frac{{{{\tan }^2}\,\alpha }}{{1 + {{\tan }^2}\,\alpha }} + \frac{{{{\tan }^2}\,\beta }}{{1 + {{\tan }^2}\,\beta }} + \frac{{{{\tan }^2}\,\gamma }}{{1 + {{\tan }^2}\,\gamma }}} \right) = 3\,{\sec ^2}\,\frac{\theta }{2},$ then $\theta =$

$2{\sin ^2}x + {\sin ^2}2x = 2,\, - \pi < x < \pi ,$ then $x = $