If $\cos (A + B) = \alpha \cos A \cos B + \beta \sin A \sin B,$ then $(\alpha, \beta) =$

If $A + B + C = \pi$ and $\cos A = \cos B \cos C$,then $\tan B \tan C$ is equal to

The value of $x,$ which satisfies the equation $2 \operatorname{cosec}^{2} 30^{\circ} + x \sin^{2} 60^{\circ} - \frac{3}{4} \tan^{2} 30^{\circ} = 10$ is

If $\cos^6 \alpha + \sin^6 \alpha + K \sin^2 2\alpha = 1,$ then $K =$

An observer on the top of a cliff,$200\, m$ above the sea level,observes the angle of depression of two ships at anchor to be $45^{\circ}$ and $30^{\circ}$ respectively. Find the distance between the ships,if the line joining them stretches to the base of the cliff. (in $m$)

If $x + \frac{1}{x} = 2\cos \alpha$, then ${x^n} + \frac{1}{{{x^n}}} = $