The product $(1+\tan 1^{\circ})(1+\tan 2^{\circ})(1+\tan 3^{\circ}) \dots (1+\tan 45^{\circ})$ equals

For $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,if $2 \cos \theta + \sin \theta = 1$ and $7 \cos \theta + 6 \sin \theta = k$,then the possible values of $k$ are:

If $\sum_{r=1}^{13} \left\{ \frac{1}{\sin \left(\frac{\pi}{4} + (r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4} + \frac{r\pi}{6}\right)} \right\} = a\sqrt{3} + b$,where $a, b \in \mathbb{Z}$,then $a^2 + b^2$ is equal to:

If $\theta+\phi=\alpha$ and $\tan \theta=k \tan \phi$ (where $k>1$),then the value of $\sin (\theta-\phi)$ is

$\operatorname{sech}^2\left(\tanh ^{-1} \frac{1}{2}\right)+\operatorname{cosech}^2\left(\operatorname{coth}^{-1} 3\right)=$

If $\sin \alpha - \cos \alpha = m$ and $\sin 2 \alpha = n - m^2$,where $-\sqrt{2} \leq m \leq \sqrt{2}$,then $n$ is equal to