Answer
$x=-\frac{1}{2}$
Work Step by Step
Use a calculator to fund the value of $arctan{(1)}$ to obtain:
$\arccos{(x)}+\frac{\pi}{4}=\frac{11\pi}{12}
\\\arccos{(x)}=\frac{11\pi}{12}-\frac{\pi}{4}
\\\arccos{(x)}=\frac{2\pi}{3}$
RECALL:
$\arccos{(x)}=\theta \longrightarrow \cos{\theta} = x$
Use the rule above to obtain:
$\arccos{(x)}=\frac{2\pi}{3}
\\\cos{(\frac{2\pi}{3})}=x
\\-\frac{1}{2}=x$
