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Question

If 0α,β90 and tan(α+β)=3 and tan(αβ)=2 then value of sin2α is ?

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Solution

Here,0α,β901stQuadrant

Let A=α+βandB=αβ

A+B=2α

Using tan(A+B)=tanA+tanB1tanAtanB

tan(2α)=tan[(α+β)+(αβ)]

tan(2α)=tan(α+β)+tan(αβ)1tan(α+β)tan(αβ)

tan(2α)=3+2132=55

tan(2α)=1<0

Now ,0α90

02α1801stor2ndQuadrant

tan(2α)=1<02ndQuadrant

2α=ππ4

sin(2α)=sin(ππ4)=sinπ4

sin(2α)=12

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