If $m \cdot \tan (\theta-30^{\circ})=n \cdot \tan (\theta+120^{\circ})$,then $\frac{m+n}{m-n}=$

If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p$,then the value of $\tan p$ is :

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

If $\theta$ is eliminated from the equations $x = a \cos(\theta - \alpha)$ and $y = b \cos(\theta - \beta)$,then $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\alpha - \beta)$ is equal to

If $\cosh \alpha + \sinh \alpha = e^3$ and $\sinh x = \frac{\alpha}{\alpha+1}$,then $\tanh x =$

The value of ${e^{\log_{10} \tan 1^\circ + \log_{10} \tan 2^\circ + \log_{10} \tan 3^\circ + \dots + \log_{10} \tan 89^\circ}}$ is