If $\cos A = \frac{\sqrt{3}}{2},$ then $\tan 3A = $

If $\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1$,then the value of $\left[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} \right]$ is (where $[.]$ denotes the greatest integer function).

The simplest value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ is

If $2\cos\theta + \sin\theta = 1$,then the value of $4\cos\theta + 3\sin\theta$ is equal to

$\frac{\cot \theta - \operatorname{cosec} \theta + 1}{\cot \theta + \operatorname{cosec} \theta - 1}$ is equal to

If $(\sin \alpha + \operatorname{cosec} \alpha)^{2} + (\cos \alpha + \sec \alpha)^{2} = K + \tan^{2} \alpha + \cot^{2} \alpha$,then the value of $K$ is