If `3sinbeta=sin(2alpha+beta)` then `tan(alpha+beta)-2tanalpha` is independent of `alpha` independent of `beta` dependent of both `alphaa n dbeta` independent of both `alphaa n dbeta`

Options

A. independent of `alpha`
B. independent of `beta`
C. dependent of both `alpha` and `beta`.
D. independent of both `alpha` and `beta`.

Answer

A, B, D

Explanation

`3 sin beta = sin ( 2alpha + beta)," "`(Given)
`tan ( alpha + beta) - 2 tanalpha `
`" " = tan (alpha + beta ) - tanalpha - tanalpha`
`" " = (sin(alpha+ beta))/( cos(alpha + beta)) - (sin alpha)/( cos alpha) - tan alpha `
`" " = ( sin(alpha + beta - alpha)) /( cos alpha cos( alpha + beta)) -tan alpha`
`" " = ( sin beta) /( cosalpha cos( alpha + beta)) - tan alpha `
`" " = ( sin beta - sin alpha cos( alpha + beta))/( cos ( alpha + beta) cos alpha) `
` " " = (2 sin beta - [ sin ( 2alpha + beta) - sin beta])/( 2cos( alpha + beta) cos alpha)`
`" " (-[sin(2alpha+ beta)- 3sin beta])/( 2cos( alpha + beta) cos alpha) =0`