Let $P_1, P_2, P_3$ be the altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively. If $\Delta$ denotes the area of the triangle and $s$ is the semi-perimeter of the triangle, then $\frac{\cos A}{P_1} + \frac{\cos B}{P_2} + \frac{\cos C}{P_3} = $

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 $\cos R > \frac{p}{q}$

The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2 \sin^2 \theta = \cos 2\theta$ and $2 \cos^2 \theta = 3 \sin \theta$ is

The number of solutions of the equation $\left( 1 - \frac{1}{2 \sin x} \right) \cos^2 2x = 2 \sin x - 3 + \frac{1}{\sin x}$ in the interval $[0, 4\pi]$ is:

In a triangle $ABC$,if $(r_1-r_3)(r_1-r_2)-2r_2r_3=0$,then $a^2-b^2=$

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