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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 (cosαcos β)+b(sin αsin β)=0

2 a sin(α+β2) sin(αβ2)+2b cos(α+β2) sin(αβ2)=0 [cos Ccos D=2 sin(C+D2) sin(CD2)andsin Csin D=2 cos(C+D2) sin(CD2)]

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(AB)]

=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(2080)]

=14[12 cos80+12(cos100+cos60)]=14[12cos80+12{cos(18080)+12}]

=14(12 cos8012 cos80+14)=116=RHS

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