यदि `2 tan alpha = 3 tan beta` हो, तो सिद्ध कीजिए - `tan (alpha - beta)=(sin 2beta)/(5-cos 2beta)`

Explanation

`2 tan alpha = 3 tan beta rArr tan alpha = 3//2 tan beta`
`L.H.S. = tan (alpha - beta)=(tan alpha - tan beta)/(1+tan alpha.tan beta)=((3)/(2)tan beta - tan beta)/(1+(3)/(2)tan beta.tan beta)`
`=(tan beta)/(2+3 tan^(2)beta) = ((sin beta)/(cos beta))/(2+3(sin^(2)beta)/(cos beta))=(sin beta cos beta)/(2cos^(2)beta + 3sin^(2)beta)`
`rArr " " L.H.S. = (2 sin beta cos beta)/(4 cos^(2)beta+6 sin^(2)beta)=(sin 2beta)/(4(cos^(2)beta+sin^(2)beta)+2sin^(2)beta)`
`=(sin 2beta)/(4+(1-cos 2beta))=(sin 2beta)/(5-cos 2beta)=R.H.S.` इति सिद्धम्