# When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?

This is a contest math question that I don't remember the reference.

When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?

Obviously, $f(0;\alpha)=0$ for all $\alpha\in{\mathbb R}$. I've no idea what theorem can be used here. Any idea?

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Not, I hope, an ongoing contest? –  Gerry Myerson Oct 10 '12 at 4:36
@GerryMyerson:No. I know the policy of math.SE and I've read this post: math.SE policy on question from ongoing contests on meta. If there is any unnecessary "confusion", I would vote to close this question. –  Jack Oct 10 '12 at 12:28

$$\cos(\alpha x) - 1 -\sin^2(x) = 0 \implies \cos(\alpha x) = 1 + \sin^2(x) \geq 1$$
$$|\cos(x)| \leq 1 \,.$$