# Use the Intermediate Value Theorem to show that cosx=x have at least a solution in [0,π]?

$\cos x - x = 0$
Now let $y = \cos x - x$. We see that $y \left(0\right) = \cos \left(0\right) - 0 = 1$ and $y \left(\pi\right) = - 1 - \pi$
Since $y \left(\pi\right) < 0 < y \left(0\right)$, and $y$ is continuous, there must be a value of $x$ in $\left[0 , \pi\right]$ where $\cos x - x = 0$.