Question

If $a^2,b^2,c^2$ are in $A.P.$, prove that $\cot A,\cot B$ and $\cot C$ are also in $A.P.$

Answer 1

$b^2-a^2=c^2-b^2$
${\sin }^{2}B-{\sin }^{2}A={\sin }^{2}C-{\sin }^{2}B$
or $\sin (B+A)\sin (B-A)$
$=\sin (C+B)\sin (C-B)$
or $\sin C(\sin B\cos A-\cos B\sin A)$
$=\sin A(\sin C\cos B-\cos C\sin B)$
Divide each term by $\sin A\sin B\sin C$
$\therefore \cot A-\cot B=\cot B-\cot C$
$\therefore \cot A,\cot B,\cot C$ are $A.P.$

Hence proved.