Fórum de Matemática
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Thanks friends i have got it

If \(Ax^2+Bx+C=0\) has more then Two Roots, Then It will become an Identity Which is True for all Real \(x\)

So \(A=B=C=0\)

Now here \(\left(a-\sin \theta\right)\alpha^2+b\alpha+\left(c+\cos \theta\right) = 0\)

Similarly \(\left(a-\sin \theta\right)\beta^2+b\beta+\left(c+\cos \theta\right) = 0\)

and \(\left(a-\sin \theta\right)\gamma^2+b\gamma+\left(c+\cos \theta\right) = 0\)

Now again If Given equation has more the 2 roots, then

\(a-\sin \theta = b = c+\cos \theta = 0\)

So \(a = \sin \theta\) and \(b = 0\) and \(c = -\cos \theta\)

Now \(a+b+c = 1\)

So \(\sin \theta-\cos \theta = 1\Leftrightarrow \sin \left(\theta -\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \sin \left(\frac{\pi}{4}\right)\)

So \(\theta = \frac{\pi}{2}\;\;,\pi\)