The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4}$. If $\sin (\alpha+\beta) = \frac{1}{3}$ and $\cos (\alpha-\beta) = \frac{2}{3}$,then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha}\right)^2$ is:

If $\sin \theta + \operatorname{cosec} \theta = 4$,then $\sin^2 \theta + \operatorname{cosec}^2 \theta = $

If $\alpha = 22^\circ 30'$,then $(1 + \cos \alpha )(1 + \cos 3\alpha )(1 + \cos 5\alpha )(1 + \cos 7\alpha )$ equals

If $\tan \left(\frac{\pi}{4}+\alpha\right)=\tan ^3\left(\frac{\pi}{4}+\beta\right)$,then $\tan (\alpha+\beta) \cot (\alpha-\beta)=$

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