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

Let $\tan \alpha, \tan \beta$ and $\tan \gamma$ (where $\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}, n \in N$) be the slopes of three line segments $OA, OB$ and $OC$ respectively,where $O$ is the origin. If the circumcentre of $\Delta ABC$ coincides with the origin and its orthocentre lies on the $y$-axis,then the value of $\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^2$ is equal to:

If $\sin x + \operatorname{cosec} x = 3$,then the value of $\sin^{4} x + \operatorname{cosec}^{4} x$ is

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

$\frac{\cos 12^{\circ}-\sin 12^{\circ}}{\cos 12^{\circ}+\sin 12^{\circ}}+\frac{\sin 147^{\circ}}{\cos 147^{\circ}} = $

If $y^2+z^2=3yz$,$z^2+x^2=8zx$,and $x^2+y^2=4xy$,then the value of $\frac{y^2}{xz}+\frac{xz}{y^2}$ is