If $\tan A = - \frac{1}{2}$ and $\tan B = - \frac{1}{3},$ then $A + B = $

$\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = $

If $\tan \alpha = \frac{x^2 - x}{x^2 - x + 1}$ and $\tan \beta = \frac{1}{2x^2 - 2x + 1}$ $(x \ne 0, 1)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\tan(\alpha + \beta)$ has the value equal to:

The value of ${e^{{{\log }_{10}}\tan 1^\circ + {{\log }_{10}}\tan 2^\circ + {{\log }_{10}}\tan 3^\circ + ........... + {{\log }_{10}}\tan 89^\circ }}$ is

If $\tan A = \frac{n}{n+1}$ and $\tan B = \frac{1}{2n+1}$,the value of $\tan (A+B) =$

If $\sec \alpha + \tan \alpha = 2,$ then the value of $\sin \alpha$ is (assume that $0 < \alpha < 90^{\circ}$).