If $A + B + C = \pi$ & $sin\, \left( {A\,\, + \,\,\frac{C}{2}} \right) = k \,sin,\frac{C}{2}$ then $tan\, \frac{A}{2} \,tan \, \frac{B}{2}=$

The sum of the solutions in $x \in (0,4\pi )$ of the equation $4\sin \frac{x}{3}\left( {\sin \left( {\frac{{\pi  + x}}{3}} \right)} \right)\sin \left( {\frac{{2\pi  + x}}{3}} \right) = 1$ is

$\sum\limits_{r = 1}^{100} {\frac{{\tan \,{2^{r - 1}}}}{{\cos \,{2^r}}}} $ is equal to

If $\alpha ,$ $\beta$ are different values of $x$ satisfying $a\cos x + b\sin x = c,$ then $\tan {\rm{ }}\left( {\frac{{\alpha + \beta }}{2}} \right) = $

If $\frac{{1 - \cos 2\theta }}{{1 + \cos 2\theta }} = 3$, then the general value of $\theta $ is

If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta,$ then $\lambda$ lies in the interval