Question

In `DeltaABC`, `a^(2),b^(2),c^(2)` are in A.P. Prove that cotA, cotB, cotC are also in A.P.

Answer 1

Let cotA, cotB, cotC are in A.P.
`rArr (cosA)/(sinA),(cosB)/(sinB)` are in A.P.
`rArr (b^(2)+c^(2)-a^(2))/(2bc.a/k),(c^(2)+a^(2)-b^(2))/(2ca.b/k), (a^(2)+b^(2)-c^(2))/(2ab.c/k)` are in A.P.
`rArr b^(2)+c^(2)-a^(2),c^(2)+a^(2)-b^(2),a^(2)+b^(2)-c^(2)` are also in A.P(Multiply each term by `(2abc)/(k))`
`rArr -2a^(2),-2b^(2),-2c^(2)` are in A.P. (Subtract `a^(2)+b^(2)+c^(2)` from each term)
`rArr a^(2),b^(2),c^(2)` are in A.P. (Divide each term by `-2`)
`therefore` cotA, cotB, cotC are in A.P. Hence Proved.