The value of $\log \tan 1^{\circ} + \log \tan 2^{\circ} + \log \tan 3^{\circ} + \ldots + \log \tan 89^{\circ}$ is equal to

If the maximum value of $y = \frac{7 + 6 \tan x - \tan^2 x}{1 + \tan^2 x}$ is $\lambda$,then the value of $\log_{\sqrt{2}}(\lambda)$ is:

If $\frac{2\sin \alpha}{1 + \cos \alpha + \sin \alpha} = y,$ then $\frac{1 - \cos \alpha + \sin \alpha}{1 + \sin \alpha} = $

$\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}$ is equal to

From two points,lying on the same horizontal line,the angles of elevation of the top of the pillar are $\theta$ and $\phi$ $(\theta < \phi)$. If the height of the pillar is $h$ $m$ and the two points lie on the same side of the pillar,then the distance between the two points is (in $m$):

The value of $\tan 7\frac{1}{2}^\circ$ is equal to