Question

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

Answer 1

$a^2,b^2,c^2$ are in AP,

$-2a^2,-2b^2,-2c^2$ are in AP,

$(a^2+b^2+c^2)-2a^2,(a^2+b^2+c^2)-2b^2,(a^2+b^2+c^2)-2c^2$ are in AP,

$b^2+c^2-a^2,a^2+c^2-b^2,a^2+b^2-c^2$ are in AP,

$\frac{b^2+c^2-a^2}{2abc},\frac{a^2+c^2-b^2}{2abc},\frac{a^2+b^2-c^2}{2abc}$ are in AP,

$\frac{k}{a}\frac{b^2+c^2-a^2}{2bc},\frac{k}{b}\frac{a^2+c^2-b^2}{2ac},\frac{k}{c}\frac{a^2+b^2-c^2}{2ab}$ are in AP,

Since $cosA=\frac{b^2+c^2-a^2}{2bc}$ and similarly $cosBandcosC$

$\frac{k}{a}cosA,\frac{k}{b}cosB,\frac{k}{c}cosC$ are in AP,

$\frac{cosA}{sinA},\frac{cosB}{sinB},\frac{cosC}{sinC}$ are in AP,

(Since $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=k$)

Hence,

$\cot A,\cot B,\cot C$ are in $AP.$