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

If $2 \sin x - \cos 2x = 1$,then $(3 - 2 \sin^2 x) = $

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

If in a triangle $ABC$,$AB=5$ units,$\angle B=\cos ^{-1}\left(\frac{3}{5}\right)$ and the radius of the circumcircle of $\triangle ABC$ is $5$ units,then the area (in sq. units) of $\triangle ABC$ is:

The set of values of $a$ for which the equation $cos\, 2x + a\, sin\, x = 2a - 7$ possesses a solution is:

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