If the reciprocals of the lengths of the sides of a $\triangle ABC$ are in harmonic progression,then its ex-radii $r_1, r_2, r_3$ are in

If the angles $A, B$ and $C$ of a triangle are in an Arithmetic Progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively,then the value of the expression $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A$ is

All possible values of $\theta \in [0, 2\pi]$ for which $\sin 2\theta + \tan 2\theta > 0$ lie in

In $\Delta ABC,$ if $2(bc \cos A + ca \cos B + ab \cos C) = $

If $\Delta$ denotes the area of triangle $ABC$,then $(b \sin C + c \sin B)(b \cos C + c \cos B) =$

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