If p and q are the roots of the equation x^2-\sin^2 \theta x-\cos^2 \theta=0, then what is the <strong>MINIMUM</strong> value of p^2+q^2?

Explanation

Sum of roots p+q = \sin^2 \theta, product pq = -\cos^2 \theta. We need to minimize p^2+q^2 = (p+q)^2 - 2pq = \sin^4 \theta + 2\cos^2 \theta. Substituting \cos^2 \theta = 1 - \sin^2 \theta, we get \sin^4 \theta - 2\sin^2 \theta + 2 = (\sin^2 \theta - 1)^2 + 1. Since 0 \leq \sin^2 \theta \leq 1, the minimum occurs when \sin^2 \theta = 1, giving a minimum value of 1.

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