If `a^2, b^2,c^2` are in A.P., then prove that `tanA ,tanB ,tanC` are in H.P.

Explanation

`a^(2), b^(2), c^(2)` are in A.P.
`rArr b^(2) - a^(2) = c^(2) - b^(2)`
`rArr sin^(2)B - sin^(2) A = sin^(2) C - sin^(2) B` (Using sine Rule)
or `sin(B + A) sin (B - A) = sin (C + B) sin (C - B)`
or `sin C (sin B cos A - cos B sin A)`
`= sin A (sin C cos B - cos C sin B)`
Dividing both sides by sin A sin B sin C, we get
`cot A - cot B = cot B - cot C`
`rArr cot A, cot B, cot C` are in A.P.
`rArr tan A, tan B, tan C` are in H.P.