Question

$Inany\Delta ABC,if{a}^2,b^2,c^2\text{are in A.P., prove that cot A, cot B and cot C are also in A.P.}$

Answer 1

$a^2,b^2,c^2\text{are in A.P.}\Rightarrow -2a^2,-2b^2,-2c^2\text{are in A.P.}\Rightarrow (a^2+b^2+c^2)-2a^2,(a^2+b^2+c^2)-2b^2,(a^2+b^2+c^2)-2c^2\text{are in A.P.}\Rightarrow (b^2+c^2-a^2),(c^2+a^2-b^2),(b^2+a^2-c^2)\text{are in A.P.}\Rightarrow \frac{(b^2+c^2-a^2)}{2abc},\frac{(c^2+a^2-b^2)}{2abc},\frac{(b^2+a^2-c^2)}{2abc}\text{are in A.P.}\Rightarrow \frac{1}{a}\frac{(b^2+c^2-a^2)}{2abc},\frac{1}{b}\frac{(c^2+a^2-b^2)}{2abc},\frac{1}{c}\frac{(b^2+a^2-c^2)}{2abc}\text{are in A.P.}\Rightarrow \frac{1}{a}cosA,\frac{1}{b}cosB,\frac{1}{C}cosC\text{are in A.P.}\Rightarrow \frac{k}{a}cosA,\frac{k}{b}cosB,\frac{k}{c}cosC\text{are in A.P.}\Rightarrow \frac{cosA}{sinA},\frac{cosB}{sinB},\frac{cosC}{sinC}\text{are in A.P.}\Rightarrow cotA,cotB,cotC\text{are in A.P.}$