If $29 \tan \theta = 31,$ then the value of $\frac{1+2 \sin \theta \cos \theta}{1-2 \sin \theta \cos \theta}$ is equal to

If $13 \sin A = 12$ where $\frac{\pi}{2} < A < \pi$ and $5 \sec B = 13$ where $\frac{3\pi}{2} < B < 2\pi$,then find the value of $5 \tan A + 3 \tan^2 B$.

$3\,\left[ {{{\sin }^4}\,\left( {\frac{{3\pi }}{2} - \alpha } \right) + {{\sin }^4}\,(3\pi + \alpha )} \right] - 2\,\left[ {{{\sin }^6}\,\left( {\frac{\pi }{2} + \alpha } \right) + {{\sin }^6}(5\pi - \alpha )} \right] = $

Minimum value of $8 \cos^2 x + 18 \sec^2 x$ for all $x \in R$ wherever it is defined,is:

$\cot (45^\circ + \theta ) \cot (45^\circ - \theta ) = $

The value of the following expression is $3(\sin^{4} \theta + \cos^{4} \theta) + 2(\sin^{6} \theta + \cos^{6} \theta) + 12\sin^{2} \theta \cos^{2} \theta$.