if,${\mathrm{a}}^2,{\mathrm{b}}^2,{\mathrm{c}}^2$ are in AP, then which of the following is also in AP ?

$\because {\mathrm{a}}^2,{\mathrm{b}}^2,{\mathrm{c}}^2$ are in AP.

$\begin{array}{r}\Rightarrow {\sin }^2 \mathrm{B}-{\sin }^2 \mathrm{A}={\sin }^2 \mathrm{C}-{\sin }^2 \mathrm{B}\\ \Rightarrow \sin \mathrm{C}(\sin \mathrm{Bcos} \mathrm{A}-\cos \mathrm{Bsin} \mathrm{A})\end{array}$

$=\sin \mathrm{A}(\sin \mathrm{Ccos} \mathrm{B}-\cos \mathrm{Csin} \mathrm{B})$

On dividing by sin $\mathrm{A}$sin B sin C, we get

$2\cot \mathrm{B}=\cot \mathrm{A}+\cot \mathrm{C}$

$\Rightarrow \cot \mathrm{A},\cot \mathrm{B},\cot \mathrm{C}$are in AP.