In a triangle $\mathrm{ABC}$ if $\frac{\sin \mathrm{A}-\sin \mathrm{C}}{\cos \mathrm{C}-\cos…
In a triangle $\mathrm{ABC}$ if $\frac{\sin \mathrm{A}-\sin \mathrm{C}}{\cos \mathrm{C}-\cos \mathrm{A}}=\cot \mathrm{B}$, then $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are in
Arithmetico - Geometric progression
Harmonic Progression
Geometric progression
Arithmetic progression
Solution
$\frac{\sin A-\sin C}{\cos C-\cos A}=\cot B$
$2 \cos \left(\frac{A+C}{2}\right) \cdot \sin \left(\frac{A-C}{2}\right)$
$2 \cdot \sin \left(\frac{A+C}{2}\right) \cdot \sin \left(\frac{A-C}{2}\right)$
$\cot \left(\frac{A+C}{2}\right)=\cot B \Rightarrow \frac{A+C}{2}=B \Rightarrow A+C=2 B$
$\therefore A, B, C$ are in A.P.