If $A, B, C, D$ are angles of a cyclic quadrilateral,then $\cos A + \cos B + \cos C + \cos D$ is equal to

In $\triangle PQR$,$\angle R = \frac{\pi}{4}$. If $\tan \left(\frac{P}{3}\right)$ and $\tan \left(\frac{Q}{3}\right)$ are the roots of the equation $ax^2 + bx + c = 0$,then:

The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

In a triangle $ABC$,angle $A$ is greater than angle $B$. If the measures of angles $A$ and $B$ satisfy the equation $3\sin x - 4\sin^3 x - k = 0$ for $0 < k < 1$,then the measure of angle $C$ is:

If $x, y, z$ are the lengths of the perpendiculars drawn from the circumcenter to the sides $a, b, c$ respectively of a triangle,then the value of $\frac{bx}{c} + \frac{cy}{a} + \frac{az}{b}$ is

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):
$(A) 16$
$(B) 18$
$(C) 24$
$(D) 22$