If $A + B + C = \pi$ & $sin\, \left( {A\,\, + \,\,\frac{C}{2}} \right) = k \,sin,\frac{C}{2}$ then $tan\, \frac{A}{2} \,tan \, \frac{B}{2}=$
$\frac{{k\,\, - \,\,1}}{{k\,\, + \,\,1}}$
$\frac{{k\,\, + \,\,1}}{{k\,\, - \,\,1}}$
$\frac{k}{{k\,\, + \,\,1}}$
$\frac{{k\,\, + \,\,1}}{k}$
The number of values of $x$ for which $sin\,\, 2x + cos\,\, 4x = 2$ is
The solution of the equation $\left| {\,\begin{array}{*{20}{c}}{\cos \theta }&{\sin \theta }&{\cos \theta }\\{ - \sin \theta }&{\cos \theta }&{\sin \theta }\\{ - \cos \theta }&{ - \sin \theta }&{\cos \theta }\end{array}\,} \right| = 0$, is
Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}$ and $\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta-14)^2$ is equal to
The solution of $3\tan (A - {15^o}) = \tan (A + {15^o})$ is
Number of values of $x$ satisfying $2sin^22x = 2cos^28x + cos10x$ in $x \in \left[ { - \frac{\pi }{4},\frac{\pi }{4}} \right]$ is-