Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, ...$. Then for all $x \in R$,the value of $f_4(x) - f_6(x)$ is equal to

If angle $\theta$ is divided into two parts such that the tangent of one part is $k$ times the tangent of the other and $\phi$ is their difference,then $\sin \theta = $

$\tan 6^{\circ} \tan 42^{\circ} \tan 66^{\circ} \tan 78^{\circ} = $

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to

If $\alpha, \beta, \gamma, \delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$,then the value of $4\sin \frac{\alpha}{2} + 3\sin \frac{\beta}{2} + 2\sin \frac{\gamma}{2} + \sin \frac{\delta}{2}$ is equal to

If $\cos x = \tan y$,$\cot y = \tan z$ and $\cot z = \tan x$,then $\sin x$ equals to