In a triangle $ABC$,if $\cos A + 2 \cos B + \cos C = 2$ and the lengths of the sides opposite to the angles $A$ and $C$ are $3$ and $7$ respectively,then $\cos A - \cos C$ is equal to

Let $S = \{x \in [-\pi, \pi] : \sin x(\sin x + \cos x) = a, a \in Z\}$. Then $n(S)$ is equal to:

In a triangle,the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$,where $c$ is the third side of the triangle,then the ratio of the in-radius to the circum-radius of the triangle is

Consider a triangle $PQR$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$,respectively. Then which of the following statements is (are) $TRUE$?
$(A)$ $\cos P \geq 1-\frac{p^2}{2qr}$
$(B)$ $\cos R \geq \left(\frac{q-r}{p+q}\right) \cos P + \left(\frac{p-r}{p+q}\right) \cos Q$
$(C)$ $\frac{q+r}{p} < 2 \frac{\sqrt{\sin Q \sin R}}{\sin P}$
$(D)$ If $p < q$ and $p < r$,then $\cos Q > \frac{p}{r}$ and $\cos R > \frac{p}{q}$

If in $\triangle ABC$,$a \tan A + b \tan B = (a + b) \tan \left(\frac{A+B}{2}\right)$,then which of the following holds?

In any triangle $ABC$,find the value of $\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2}$.