यदि (If) `2 tanalpha= 3 tan beta `,साबित करें कि
`tan ( alpha - beta) = ( sin 2 beta)/( 5-cos 2beta)`

Explanation

दिया हुआ है `2 tan alpha = 3 tan beta ` `:. tan alpha = (3)/(2) tan beta `
अब ` 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 )` ...(i)
`= ( tan beta)/( 2+ 3 tan^(2) beta) = ((sin beta )/(cos beta))/(2+3(sin^(2) beta)/(cos^(2) beta))= ( sin betacos beta)/(2 cos^(2) beta + 3 sin^(2) beta )`
`=(sin beta cos beta)/( 1+ cos 2 beta+ 3. (1)/(2) ( 1- cos 2 beta))= ( 2 sin beta. cos beta)/( 5+2 cos 2beta - 3 cos 2 beta)`
`= ( sin 2 beta)/( 5-cos 2 beta)`