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If tan (alpha-beta)=(sin 2beta)/(3-cos 2...

If `tan (alpha-beta)=(sin 2beta)/(3-cos 2beta)`, then

A

`tan alpha=2 tan beta`

B

`tan beta=2 tan alpha`

C

`2 tan alpha=3 tan beta`

D

`3 tan alpha=2 tan beta`

Text Solution

Verified by Experts

The correct Answer is:
A
We have `(sin 2beta)/(2-cos 2beta)=(2sin beta-cos beta)/(2-2cos2beta+1+cos2beta)`
`=(2sin beta cos beta)/(4sin^(2)beta+2cos^(2)beta)`
`=(tan beta)/(1+2tan^(2)beta)`
`=(2tan beta-tanbeta)/(1+2tan^(2)beta)`
`=tan (alpha-beta)`.
`therefore (tan alpha-tan beta)/(1+tan alpha-tan beta)=(2tan beta-tan beta)/(1+2tan^(2)beta)`
`therefore tan alpha=2 tan beta`
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