Let $A, B$ and $C$ be the angles of a triangle and $\tan \frac{A}{2}=\frac{2}{5}, \tan \frac{B}{2}=\frac{3}{7}$. Then, $\tan \frac{C}{2}$ is equal to

Let $\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}$, where $A, B$ and $C$ are angles of a $\triangle A B C$. If the lengths of the sides opposite these angles are $a, b$ and $c$ respectively, then

Let $\alpha$ be the solution of ${16^{{{\sin }^2}\theta }} + {16^{{{\cos }^2}\theta }} = 10$ in $\left( {0,{\pi \over 4}} \right)$. If the shadow of a vertical pole is ${1 \over {\sqrt 3 }}$ of its height, then the altitude of the sun is

Given that a house forms a right angle view from a window of another house, and the angle of elevation from the base of the first house to the window is 60 degrees. If the separation between the two houses is 6 meters, calculate the height of the first house.

If in a $\Delta$ABC, 2b² = a² + c², then $\frac{\sin 3B}{\sin B}$ is equal to

The height of the chimney when it is found that on walking towards it 50 m in the horizontal line through its base, the angle of elevation of its top changes from 30$^\circ$ to 60$^\circ$ is :

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45$^\circ$. If flies off horizontally straight way from the point O. After one second, the elevation of the bird from O is reduced to 30$^\circ$, then the speed (in m/s) of the bird is