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If tanbeta=(tanalpha+tangamma)/(1+tanalp...

If `tanbeta=(tanalpha+tangamma)/(1+tanalphatangamma)dot` prove that `s in2beta=(sin2alpha+sin2gamma)/(1+sin2alphasin2gamma)` .

Text Solution

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`tan beta=((sin alpha)/(cos alpha)+(sin gamma)/(cos gamma))/(1+(sin alpha)/(cos alpha)(sin gamma)/(cos gamma))=(sin (alpha+gamma))/(cos (alpha-gamma))`
`sin2beta=(2 tan beta)/(1+tan ^(2)beta)=(2(sin(alpha+gamma))/(cos (alpha-gamma)))/(1+(sin^(2)(alpha+gamma))/(cos^(2)(alpha-gamma)))`
`=(2sin(alpha+gamma)cos(alpha-gamma))/(cos^(2)(alpha-gamma)+sin^(2)(alpha+gamma))`
`=(sin2 alpha+sin 2gamma))/(1+cos(2)(alpha-gamma))/(2)+(1-cos2(alpha+gamma))/(2))`
`(sin 2 alpha+sin 2gamma)/(1+(1)/(2)xx2 sin 2alphasin2gamma)=(sin 2 alpha+sin 2 gamma)/(1+sin2alpha sin 2 gamma)`
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