In any `triangle ABC` , if `a^2,b^2,c^2` are in AP then that cot A , cot B , cot C are in are in A.P.

Explanation

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