An angle θ is divide into two parts α,β such that tanα:tanβ=x:y then prove that sin(α-β)={(x-y)/(x+y)}sinθ

Steps to prove sin(α-β)={(x-y)/(x+y)}sinθ

From Question We know that 

⇒tanα:tanβ=x:y

⇒tanα/tanβ=x/y

⇒(sinα/cosα)/(sinβ/cosβ)=x/y

⇒(sinα.cosβ)/(cosβ.sinα)=x/y

By componendo and dividendo 

⇒{(sinα.cosβ)+(cosα.sinβ)}/{(cosβ.sinα)-(cosα.sinβ)}=(x+y)/(x-y)

⇒sin(α+β)/sin(α-β)=(x+y)/(x-y)

⇒sinθ/sin(α-β)=(x+y)/(x-y)

⇒sin(α-β)={(x-y)/(x+y)}sinθ

Hence sin(α-β)={(x-y)/(x+y)}sinθ is proved.

Detail Information:-

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