If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

If $0 < \theta < \frac{\pi}{4}$ and $8 \cos \theta + 15 \sin \theta = 15$,then $15 \cos \theta - 8 \sin \theta = $

The smallest integer $n$ such that $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \sin 48^{\circ}}+\ldots+\frac{1}{\sin 133^{\circ} \sin 134^{\circ}}=\frac{1}{\sin \left(n^{\circ}\right)}$ is

If $\tan \beta = \frac{n \sin \alpha \cos \alpha}{1 - n \cos^2 \alpha}$,then $\tan (\alpha + \beta) \cdot \cot \alpha =$

If $1+\sqrt{1+a}=(1+\sqrt{1-a}) \cot \alpha$ and $0 < a < 1$,then $\sin 4 \alpha=$

If $\cosh \beta = \sec \alpha \cos \theta$ and $\sinh \beta = \operatorname{cosec} \alpha \sin \theta$,then $\sinh^2 \beta =$