If `a^(2),b^(2),c^(2)` are in AP, then which of the following is also an AP?

Options

A. sin A,sin B , sin c
B. tan A, tan B tan C
C. cot A , cot B , cot C
D. None of these

Answer

C

Explanation

`sin^(2) B - sin^(2)A = sin^(2) C - sin^(2)B " "[because a^(2), b^(2),c^(2) "are in AP"]`
`rArr sin (B + A) sin (B-A) = sin (C+B) sin(C -B)`
` rArr sin C(sin B cos A A - cos B sinA)`
` = sin A( sin C cos B - cos C sinA)" " [because A+B+C = pi]`
`rArr 2 cot B = cot A + cot C" "["divide by sin A sin B sin C"]`
Hence, cot A cot B and cot C are in AP.