If the two angles on the base of a triangle are $22.5^o$ and $112.5^o$,then the ratio of the height of the triangle to the length of the base is

In $\Delta ABC$,the expression $a(\cos^2 B + \cos^2 C) + \cos A(c \cos C + b \cos B)$ is equal to:

In a triangle $ABC$,$\angle B = \frac{\pi}{3}$ and $\angle C = \frac{\pi}{4}$,and $D$ divides $BC$ internally in the ratio $1 : 3$. Then $\frac{\sin \angle BAD}{\sin \angle CAD}$ is equal to

In $\triangle ABC$,find the value of $r r_1 \cot \frac{A}{2} + r r_2 \cot \frac{B}{2} + r r_3 \cot \frac{C}{2}$.

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

Let $PQR$ be a triangle of area $\Delta$ with $a=2, b=\frac{7}{2}$ and $c=\frac{5}{2}$,where $a, b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P, Q$ and $R$ respectively. Then $\frac{2 \sin P-\sin 2P}{2 \sin P+\sin 2P}$ equals