If $\cos x + \cos y + \cos \alpha = 0$ and $\sin x + \sin y + \sin \alpha = 0,$ then $\cot \left( \frac{x + y}{2} \right) = $

If $P = \sin \frac{2 \pi}{7} + \sin \frac{4 \pi}{7} + \sin \frac{8 \pi}{7}$ and $Q = \cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{8 \pi}{7}$,then the point $(P, Q)$ lies on the circle of radius

The smallest positive value of $x$ (in degrees) for which $\tan(x+100^{\circ}) = \tan(x+50^{\circ}) \tan(x) \tan(x-50^{\circ})$ is (in $^{\circ}$)

If $\cos A+\cos (A+B)+\cos (A+2 B)+\ldots$ up to $n$ terms $=$ $\cos \left(\frac{2 A+(n-1) B}{2}\right) \sin \frac{n B}{2} \operatorname{cosec} \frac{B}{2}$,then $\cos \frac{\pi}{19}+\cos \frac{3 \pi}{19}+\cos \frac{5 \pi}{19}+\ldots+\cos \frac{17 \pi}{19} = $

If $\sin x + \sin y = p$ and $\cos x + \cos y = q$,then $\sec(x + y) = $

If $\sin x - \sin y = \frac{27}{65}$ and $\cos x - \cos y = \frac{-21}{65}$,then find the value of $\sin(x + y)$.