If $x = \frac{n\pi}{2}$ satisfies the equation $\sin \frac{x}{2} - \cos \frac{x}{2} = 1 - \sin x$ and the inequality $\left| \frac{x}{2} - \frac{\pi}{2} \right| \le \frac{3\pi}{4}$,then:

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

$A$ triangle $ABC$ has area of $P$ square units and perimeter $2S$ units. If $h_1, h_2$ and $h_3$ are respectively the lengths of the altitudes of the triangle drawn from the vertices $A, B$ and $C$,then $P^2 \left[ \frac{(h_1 h_2 + h_2 h_3 + h_3 h_1)^2}{h_1^2 h_2^2 h_3^2} - 2 \right] =$

If the equation $2 \sin^2 x + \frac{\sin 2x}{2} = k$ has at least one real solution,then the sum of all integral values of $k$ is

In a $\Delta ABC$,if $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$ and the side $a = 2$,then the area of the triangle is:

If $A$ is the solution set of the equation $\cos ^2 x = \cos ^2 \frac{\pi}{6}$ and $B$ is the solution set of the equation $\cos ^2 x = \log _{16} P$ where $P + \frac{16}{P} = 10$,then $B - A =$