The smallest positive root of the equation $tanx\,  -\,  x = 0$ lies on

Let $P = \left\{ {\theta :\sin \,\theta  - \cos \,\theta  = \sqrt 2 \,\cos \,\theta } \right\}$ and $Q = \left\{ {\theta :\sin \,\theta  + \cos \,\theta  = \sqrt {2\,} \sin \,\theta } \right\}$ be two sets. Then

Solve $\cos x=\frac{1}{2}$

Find the principal solutions of the equation $\sin x=\frac{\sqrt{3}}{2}$

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ 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 $\theta =$