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Question

# If α and β are distinct roots of a cos θ+b sin θ=c, Prove that sin(α+β)=2aba2+b2. or Prove that cos 20∘ cos40∘ cos60∘ cos80∘=116

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Solution

## It is given that, α and β are the distinct roots of a cosθ+b sinθ=c ∴a cosα+b sinα=c ...(i) and a cosβ+b sin β=c ...(ii) From Eqs. (i) and (ii), we get a (cosα−cos β)+b(sin α−sin β)=0 ⇒−2 a sin(α+β2) sin(α−β2)+2b cos(α+β2) sin(α−β2)=0 [∵cos C−cos D=−2 sin(C+D2) sin(CD2)andsin C−sin D=2 cos(C+D2) sin(C−D2)] ⇒tan α+β2=ba Now, sin (α+β)=2 tan(α+β2)1+tan2(α+β2)=2ba1+b2a2=2aba2+b2 We have, LHS=cos20∘ cos40∘ cos60∘ cos80∘=cos60∘ cos20∘ cos40∘ cos80∘ =12(cos 20∘ cos40∘) cos80∘ =14(2 cos20∘ cos40∘) cos80∘ [∵ 2cos A cos B=cos(A+B)+cos(A−B)] =14(cos 60∘+cos 20∘)cos 80∘=14(cos 60∘ cos80∘+cos 20∘ cos80∘) =14(12 cos80∘+12.2 cos20∘.cos80∘) =14[12cos 80∘+12cos(20∘+80∘)+cos(20∘−80∘)] =14[12 cos80∘+12(cos100∘+cos60∘)]=14[12cos80∘+12{cos(180−80∘)+12}] =14(12 cos80∘−12 cos80∘+14)=116=RHS

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