In $\triangle ABC$,if $a, b$ and $c$ are in arithmetic progression,then $\cos A + 2 \cos B + \cos C =$

The angles of depression of the top and the foot of a chimney as seen from the top of a second chimney,which is $150 \, m$ high and standing on the same level as the first,are $\theta$ and $\phi$ respectively. If $\tan \theta = \frac{4}{3}$ and $\tan \phi = \frac{5}{2}$,then the distance between their tops is.......$m$.

If $x = \frac{n\pi}{2}$ satisfies the equation $\sin \frac{x}{2} - \cos \frac{x}{2} = 1 - \sin x$ and the inequality $\left| \frac{x}{2} - \frac{\pi}{2} \right| \le \frac{3\pi}{4}$,then:

In a non-right-angled triangle $\triangle PQR$, let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $PQ$ at $S$, the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $O$. If $p=\sqrt{3}, q=1$, and the radius of the circumcircle of the $\triangle PQR$ equals $1$, then which of the following options is/are correct?
$(1)$ Area of $\triangle SOE = \frac{\sqrt{3}}{48}$
$(2)$ Radius of incircle of $\triangle PQR = \frac{\sqrt{3}}{2}(2-\sqrt{3})$
$(3)$ Length of $RS = \frac{\sqrt{7}}{2}$
$(4)$ Length of $OE = \frac{1}{6}$

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

In $\Delta ABC$,if $\sin A : \sin C = \sin (A - B) : \sin (B - C)$,then