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

If $10 \sin^4 \theta + 15 \cos^4 \theta = 6$,then the value of $\frac{27 \operatorname{cosec}^6 \theta + 8 \sec^6 \theta}{16 \sec^8 \theta}$ is:

If $x \sin^3 \alpha + y \cos^3 \alpha = \sin \alpha \cos \alpha$ and $x \sin \alpha - y \cos \alpha = 0$,then $x^2 + y^2 = $

If $\cos \alpha + \cos \beta = a$,$\sin \alpha + \sin \beta = b$ and $\alpha - \beta = 2 \theta$,then $\frac{\cos 3 \theta}{\cos \theta} = $

If $\cosh \alpha + \sinh \alpha = e^3$ and $\sinh x = \frac{\alpha}{\alpha+1}$,then $\tanh x =$

Let the set of all $a \in \mathbb{R}$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9^{\circ} - \tan 27^{\circ} - \frac{1}{\cot 63^{\circ}} + \tan 81^{\circ}$,then $pqr$ is equal to ....................