For $0 < \theta < \frac{\pi}{2}$,the solution$(s)$ of $\sum_{m=1}^6 \operatorname{cosec}\left(\theta+\frac{(m-1) \pi}{4}\right) \operatorname{cosec}\left(\theta+\frac{m \pi}{4}\right) = 4 \sqrt{2}$ is(are):

$\frac{\sin 3A - \cos \left( \frac{\pi}{2} - A \right)}{\cos A + \cos (\pi + 3A)} = $

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:

$\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ}=$

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

If $y = (1 + \tan A)(1 - \tan B)$ where $A - B = \frac{\pi}{4}$,then $(y + 1)^{y + 1}$ is equal to