Question

In any $\Delta ABC$, if $a^2,b^2,c^2$ are in arithmetic progression, then prove that $\cot A,\cot B,\cot C$ are in arithmetic progression.

Answer 1

Given that $a^2,b^2,c^2$ are in arithmetic progression.

We need to prove that $\cot A,\cot B$ and $\cot C$ are in A.P.
$\Rightarrow a^2,b^2,c^2$ are in A.P.
Therefore, $-2a^2,-2b^2,-2c^2$ are in A.P.
$\Rightarrow (a^2+b^2+c^2)-2a^2,(a^2+b^2+c^2)-2b^2,(a^2+b^2+c^2)-2c^2$ are in A.P.
$\Rightarrow (b^2+c^2-a^2),(c^2+a^2-b^2),(a^2+b^2-c^2)$ are in A.P.
$\Rightarrow \frac{(b^2+c^2-a^2)}{2abc},\frac{(c^2+a^2-b^2)}{2abc},\frac{(a^2+b^2-c^2)}{2abc}$ are in A.P.
$\Rightarrow \frac{1}{a}\left(\frac{b^2+c^2-a^2}{2bc}\right),\frac{1}{b}\left(\frac{c^2+a^2-b^2}{2ac}\right),\frac{1}{c}\left(\frac{a^2+b^2-c^2}{2ab}\right)$ are in A.P.
$\Rightarrow \frac{1}{a}\cos A,\frac{1}{b}\cos B,\frac{1}{c}\cos C$ are in A.P.
$\Rightarrow \frac{K}{a}\cos A,\frac{K}{b}\cos B,\frac{K}{c}\cos C$ are in A.P.
$\Rightarrow \frac{\cos A}{\sin A},\frac{\cos B}{\sin B},\frac{\cos C}{\sin C}$ are in A.P.
$\Rightarrow \cot A,\cot B,\cot C$ are in A.P.