Question

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)

Answer 1

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 : $p\in(0,\pi)$