In $\triangle ABC$,if $A = 60^{\circ}$ and $B = 105^{\circ}$,then find the value of $\frac{2R^2(b-c) \sin A \sin B \sin C}{(b+c)(s-a \cos C - c \cos A)(s-a \cos B - b \cos A)}$.

If in $\Delta ABC,$ $a = 6, b = 3$ and $\cos(A - B) = \frac{4}{5},$ then its area will be ..... $square \, unit.$

Let in a right-angled triangle,the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right-angled triangle,then $\sin \theta$ is equal to:

Assertion $(A)$: If $A=15^{\circ}, B=17^{\circ}$ and $C=13^{\circ}$,then $\cot 2A + \cot 2B + \cot 2C = \cot 2A \cot 2B \cot 2C$.
Reason $(R)$: In a $\triangle PQR$,$\tan \frac{P}{2} \tan \frac{Q}{2} + \tan \frac{Q}{2} \tan \frac{R}{2} + \tan \frac{P}{2} \tan \frac{R}{2} = 1$.
The correct option among the following is:

In $\Delta ABC$,$(a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = $

Let $P_1, P_2, P_3$ be the altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively. If $\Delta$ denotes the area of the triangle and $s$ is the semi-perimeter of the triangle, then $\frac{\cos A}{P_1} + \frac{\cos B}{P_2} + \frac{\cos C}{P_3} = $