In a triangle $ABC,$ ${a^3}\cos (B - C) + {b^3}\cos (C - A) + {c^3}\cos (A - B) = $

In a $\triangle ABC$,with usual notation,match the items in List-$I$ with the items in List-$II$ and choose the correct option.

List-$I$	List-$II$
$(A) \ r_1 r_2 \sqrt{\frac{4R-r_1-r_2}{r_1+r_2}}$	$1. \ b$
$(B) \ \frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}$	$2. \ a^2, b^2, c^2 \text{ are in } AP$
$(C) \ \frac{a}{c} = \frac{\sin(A-B)}{\sin(B-C)}$	$3. \ \Delta$
$(D) \ bc \cos^2 \frac{A}{2}$	$4. \ R r_1 r_2 r_3$
	$5. \ s(s-a)$

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 a triangle $ABC$,$\tan \frac{A}{2} = \frac{5}{6}$ and $\tan \frac{C}{2} = \frac{2}{5}$,then

In $\triangle ABC$,with usual notations,if $a, b, c$ are in $A.P.$,then $a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right) = $

$A$ tower is situated on a horizontal plane. Two points lie on the line passing through the base of the tower,at distances $a$ and $b$ from the base. The angles of elevation of the top of the tower from these points are $\alpha$ and $90^\circ - \alpha$. If the line segment joining the two points subtends an angle $\theta$ at the top of the tower,find the height of the tower.