If tan ( alpha + beta - gamma ) / tan ( alpha - beta + gamma ) = tan - Maths - Trigonometric Functions

Question

If tan ( alpha + beta - gamma ) / tan ( alpha - beta + gamma ) =
tan gamma / tan beta then prove that sin ( beta- gamma) =0 or sin 2
alpha + sin 2 beta+ sin 2 gamma =0

please answer this question only and please don't link to any other
question Thank you

Anuradha Sharma · Accepted Answer

tanα+β-γtanα-β+γ=tanγtanβsinα+β-γcosα+β-γsinα-β+γcosα-β+γ=sinγcosγsinβcosβsinα+β-γcosα-β+γcosα+β-γsinα-β+γ=sinγcosβcosγsinβUsing componendo and divindo, sinα+β-γcosα-β+γ+cosα+β-γsinα-β+γsinα+β-γcosα-β+γ-cosα+β-γsinα-β+γ=sinγcosβ+cosγsinβsinγcosβ-cosγsinβsinα+β-γ+α-β+γsinα+β-γ-α+β-γ=sinγ+βsinγ-βsin2αsin2β-γ=sinγ+βsinγ-βsin2α2sinβ-γcosβ-γ=sinγ+βsinγ-β-sinβ-γsin2α=2sinβ-γcosβ-γsinγ+βsinβ-γsin2α+2cosβ-γsinγ+β=0So either sinβ-γ=0or sin2α+2sinγ+βcosβ-γ=0sin2α+2sinγ+βcosβ-γ=0Using the formula sinC+sinD = 2sinC+D2cosC-D2sin2α+sin2β+sin2γ=0

If tan ( alpha + beta - gamma ) / tan ( alpha - beta + gamma ) = tan gamma / tan beta then prove that sin ( beta- gamma) =0 or sin 2 alpha + sin 2 beta+ sin 2 gamma =0 please answer this question only and please don't link to any other question. Thank you

If tan ( alpha + beta - gamma ) / tan ( alpha - beta + gamma ) = tan gamma / tan beta then prove that sin ( beta- gamma) =0 or sin 2 alpha + sin 2 beta+ sin 2 gamma =0

please answer this question only and please don't link to any other question. Thank you