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If the equation (cos p - 1 ) x^2 + (cos p)x + sin p = 0 in the variable x has real roots, then prove that p can take any value in the interval (0,π)




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Discriminant muse be nonnegative for the given equation to have real roots :
\cos^2p-4(\cos p-1)\sin~p\ge~0

Observe that \cos^2p is always nonnegative because it is square of a real number.
Also \cos p-1\le0 as |\cos p|\le1

Together imply the discriminant is nonnegative if \sin p\ge0
That means when p is in Ist or IInd quadrants, the discriminant is nonnegative : p\in(0,\pi)
0 0 0
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