# Question #3ded9

Apr 1, 2017

Let $f \left(x\right) = x - \cos x$.

We know that $f$ is continuous on $\left[0 , \frac{\pi}{2}\right]$ since it is the difference of two continuous functions $x$ and $\cos x$, and

$f \left(0\right) = - 1 < 0 < \frac{\pi}{2} = f \left(\frac{\pi}{2}\right)$.

By Intermediate Value Theorem, there exists $c \in \left(0 , \frac{\pi}{2}\right)$ s.t.
$f \left(c\right) = c - \cos \left(c\right) = 0$, which means that $c = \cos \left(c\right)$.

Hence, $x = \cos x$ has a solution $c \in \left(0 , \frac{\pi}{2}\right)$.