The expression $\frac{{{{\tan }^2}20^\circ  - {{\sin }^2}20^\circ }}{{{{\tan }^2}20^\circ \,\cdot\,{{\sin }^2}20^\circ }}$ simplifies to

Suppose $\theta $ and $\phi  (\ne 0)$ are such that $sec\,(\theta  + \phi ),$ $sec\,\theta $ and $sec\,(\theta  - \phi )$ are in $A.P.$ If $cos\,\theta  = k\,cos\,( \frac {\phi }{2})$ for some $k,$ then $k$ is equal to

If $\sin x + \cos x = \frac{1}{5},$ then $\tan 2x$ is

If $\tan \alpha = \frac{1}{7},\;\tan \beta = \frac{1}{3},$ then $\cos 2\alpha = $

If $\theta $ is an acute angle and $\sin \frac{\theta }{2} = \sqrt {\frac{{x - 1}}{{2x}}} $, then $\tan \theta $ is equal to

The sum of all values of $\theta \, \in \,\left( {0,\frac{\pi }{2}} \right)$ satisfying ${\sin ^2}\,2\theta  + {\cos ^4}\,2\theta  = \frac{3}{4}$ is