Existence of two solutions I am having a problem with the following exercise.
I need to show the $x^2 = \cos x $ has two solutions.
Thank you in advance.

Let $f(x)=x^2-\cos x$. Note that the curve $y=f(x)$ is symmetric about the $y$-axis. It will thus be enough to show that $f(x)=0$ has a unique positive solution. That there is a unique negative solution follows by symmetry. 
There is a positive solution, since $f(0)\lt 0$ and $f(100)\gt 0$. (Then use the Intermediate Value Theorem.)
For uniqueness of the positive solution, note that $f'(x)=2x+\sin x$.   In the interval $(0,\pi/2)$, $f'(x)$ is positive because both terms are positive. And for $x\ge \pi/2$, we have $f'(x)\ge \pi-1$.
