Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
up vote 3 down vote accepted


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

share|cite|improve this answer
Indulging in calculus for so long that I didn't notice how few facts I actually need to solve this "high school" problem. – Jack Oct 10 '12 at 12:34

Hint Check this fact

$$ |\cos(x)| \leq 1 \,. $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.