If `(2tan^(2)theta_(1)tan^(2)theta_(2)tan^(2)theta_(3)+tan^(2)theta_(1)tan^(2)theta_(2)+tan^(2)theta_(2)tan^(2)theta_(3)+tan^(2)theta_(3)tan^(2)theta_(1) = 1` then which of the following relations hold good ?

Options

A. `sin^(2)theta_(1) + sin^(2)theta_(2) + sin^(2)theta_(3) = 1`
B. `cos2theta_(1) + cos2theta_(2) + cios2theta_(3) = 1`
C. `sin^(2)theta_(1) + sin^(2)theta_(2) + sin^(2)theta_(3) = 2`
D. `cos2theta_(1) + cos2theta_(2) + cos2theta_(3) = -1`

Answer

A, B

Explanation

divide by `tan^(2)theta_(1)tan^(2)thetatan^(2)theta^(3)`
`rArr 2 + cot^(2)theta_(3) + cot^(2)theta_(1) + cot^(2)theta_(2) = cot^(2)theta_(1)cot^(2)theta_(2)cot^(2)theta_(3)`
`rArr cosec^(2)theta_(1) + cosec^(2)theta_(2) + cosec^(2)theta_(3)-1`
`=(cosec^(2)theta_(1)-1)(cosec^(2)theta_(2)-1)(cosec^(2)theta_(3)-1)`
`rArr cosec^(2)theta_(1) cosec^(2)theta_(2)+cosec^(2)theta_(2)cosec^(2)theta_(3)+cosec^(2)theta_(3)cosec^(2)theta_(1)`
`= cosec^(2)theta_(1)cosec^(2)theta_(2)cosec^(2)theta_(3)`
`sin^(2)theta_(1)+sin^(2)theta_(2)+sin^(2)theta_(3) = 1` (A)
of `cos2theta_(1)+cos2theta_(2)+cos2theta_(3) = 1` (B)