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Question

# If tan α, tan β are the roots of the equation x2 + ax + b = 0 then the value of sin2(α+β)+a sin(α+β)cos(α+β)+b cos2(α+β)=

A

a

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B

b

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C

ab

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D

1a+b

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Solution

## The correct option is B b tan α+tan β=−a, tan α tan β=btan (α+β)=tan α+tan β1−tan α tan βtan (α+β)=−a1−b=ab−1 Now sin2(α+β)+a sin (α+β)cos(α+β)+b cos2(α+β)=cos2(α+β)[tan2(α+β)+a tan(α+β)+b]=tan2(α+β)+a tan(α+β)+bsec2(α+β)=tan2(α+β)+a tan (α+β)+b1+tan2(α+β) =a2(b−1)2+a.ab−1+b1+a2(b−1)2=ab−1(ab−1+a)+b1+a2(b−1)2=ab−1(a+ab−ab−1)+b1+a2(b−1)2=b

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