The value of $\tan 1^{\circ} \tan 2^{\circ} \ldots \tan 89^{\circ}$ is

$\tan \left( \frac{\pi }{4} + \theta \right) - \tan \left( \frac{\pi }{4} - \theta \right) = $

If $\sin A = \frac{1}{\sqrt{10}}$ and $\sin B = \frac{1}{\sqrt{5}}$,where $A$ and $B$ are positive acute angles,then $A + B = $

Let $0 < x < \frac{\pi }{4}.$ Then $\sec 2x - \tan 2x = $

If $\theta$ is in the first quadrant and $\tan \theta = \frac{3}{4},$ then $\frac{\tan \left(\frac{\pi}{2}-\theta\right)-\sin (\pi-\theta)}{\sin \left(\frac{3 \pi}{2}+\theta\right)-\cot (2 \pi-\theta)} = $

Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ where $x \in R$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ is equal to: