In ` A B C ,` if `sin^3theta=sin(A-theta)sin(B-theta)sin(C-theta)` , then prove that `cottheta=cot A+cot B+cotCdot`

Explanation

Let `cot theta=cotA+cotB+cot C`
or `cot theta-cotA=cot B+cotC`
or `(sin(A-theta))/(sinA sin theta)=(sin (B+C))/(sinB sin C)`
or `sin(A-theta)=(sin^(2)Asintheta)/(sin B sin C)`
similarly,
`sin(B-theta)=(sin^(2)B sin theta)/(sin A sin C)`
and `sin(C-theta)=(sin^(2)C sin theta)/(sin A sin B)`
By multiplying corresponding sides of Eqs. (i),(ii) and (iii) we have
`sin^(3)theta=sin(A-theta)sin(B-theta)sin(C-theta)`.