The value of $\cos 20^{\circ} + 2 \sin^2 55^{\circ} - \sqrt{2} \sin 65^{\circ}$ is

$\tan 2 \alpha \cdot \tan \left(30^{\circ}-\alpha\right)+\tan 2 \alpha \cdot \tan \left(60^{\circ}-\alpha\right)+\tan \left(60^{\circ}-\alpha\right) \cdot \tan \left(30^{\circ}-\alpha\right)$ is equal to

If $3 \sin^4 x + 2 \cos^4 x = \frac{6}{5}$ and $x$ is an acute angle,then $\tan 2x =$

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:

Let $A$ and $B$ denote the statements:
$A: \cos \alpha + \cos \beta + \cos \gamma = 0$
$B: \sin \alpha + \sin \beta + \sin \gamma = 0$
If $\cos (\alpha - \beta) + \cos (\beta - \gamma) + \cos (\gamma - \alpha) = -\frac{3}{2}$,then:

For $0 < x \leq \pi$, $\sinh ^{-1}(\cot x)$ is equal to