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

In any triangle $ABC$,${\sin ^2}\frac{A}{2} + {\sin ^2}\frac{B}{2} + {\sin ^2}\frac{C}{2}$ is equal to:

Let $p_1, p_2, p_3$ be the altitudes of a triangle $ABC$ drawn through the vertices $A, B, C$ respectively. If $r_1=4, r_2=6, r_3=12$ are the ex-radii of triangle $ABC$, then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2}=$

With usual notations, in $\triangle ABC$, if $2a^2 = b^2 + c^2$, then $\frac{\cos 3A}{\cos A} + 2 = $

If $\alpha$ and $\beta$ are different values of $x$ satisfying $a \cos x + b \sin x = c,$ then $\tan \left( \frac{\alpha + \beta}{2} \right) = $

In a triangle $PQR$,$P$ is the largest angle and $\cos P = \frac{1}{3}$. Further,the incircle of the triangle touches the sides $PQ, QR$ and $RP$ at $N, L$ and $M$ respectively,such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then the possible length$(s)$ of the side$(s)$ of the triangle is (are):
$(A) 16$
$(B) 18$
$(C) 24$
$(D) 22$