If $\cot x=\frac{5}{12}$ for some $x \in\left(\pi, \frac{3 \pi}{2}\right)$, then $\sin 7 x\left(\cos \frac{13 x}{2}+\sin \frac{13 x}{2}\right)+\cos 7 x\left(\cos \frac{13 x}{2}-\sin \frac{13 x}{2}\right)$ is equal to

Let $\frac{\pi}{2}<\theta<\pi$ and $\cot \theta=-\frac{1}{2 \sqrt{2}}$. Then the value of

$$ \sin \left(\frac{15 \theta}{2}\right)(\cos 8 \theta+\sin 8 \theta)+\cos \left(\frac{15 \theta}{2}\right)(\cos 8 \theta-\sin 8 \theta) $$

is equal to :

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to :

If for $\theta \in\left[-\frac{\pi}{3}, 0\right]$, the points $(x, y)=\left(3 \tan \left(\theta+\frac{\pi}{3}\right), 2 \tan \left(\theta+\frac{\pi}{6}\right)\right)$ lie on $x y+\alpha x+\beta y+\gamma=0$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :