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Question

If tanα and tanβ are the roots of x2px+q=0, then find the value of sin2(α+β).

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Solution

Given, x2px+q=0

Let tanα and tanβ be the roots of the equation
Sum of the roots=tanα+tanβ=p
Product of the roots=tanαtanβ=q

We know that tan(α+β)=tanα+tanβ1tanαtanβ=p1q

From the right angled triangle,
tan(α+β)=p1q
oppositeside=p and adjacentside=1q

Hence hypotenuse=p2+(1q)2=p2+q2+12q

Now,sin(α+β)=oppositesidehypotenuse=pp2+q2+12q
sin2(α+β)=p2p2+q2+12q

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