#### Question

In any ΔABC if a^{2} , b^{2} , c^{2} are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

#### Solution

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

We need to prove that cotA, cotB and cotC are in

arithmetic progression.

a^{2} ,b^{2} ,c^{2} are in A.P.

`-2a^2, -2b^2, -2c^2 " are in A.P"`

`(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"`

`(b^2+c^2-a^2), (c^2+a^2-b^2),(a^2+b^2-c^2) " are in A.P "`

`(b^2+c^2-a^2)/(2abc), (c^2+a^2-b^2)/(2abc),(a^2+b^2-c^2) /(2abc)" are in A.P "`

`1/a(b^2+c^2-a^2)/(2bc), 1/b(c^2+a^2-b^2)/(2ac),1/c(a^2+b^2-c^2) /(2ab)" are in A.P "`

`1/acosA,1/bcosB,1/c cos C " are in A.P"`

`k/acosA,k/bcosB,k/c cos C " are in A.P"`

`cosA/sinA,cosB/sinB,cosC/sinC " are in A.P"`

`cotA,cotB,cotC " are in A.P"`