# What is cos ^ -1 ( cos (7 pi/4) )?

$\frac{7 \pi}{4}$
$\cos$ and ${\cos}^{-} 1$ are effectively just opposite functions of each other. In other words, if $\cos \left(\theta\right) = x$ then ${\cos}^{-} 1 \left(x\right) = \theta$. That means if you take the ${\cos}^{-} 1$ of a $\cos \left(\theta\right)$ statement, you just get the original $\theta$ value back.
${\cos}^{-} 1 \left(\cos \left(\theta\right)\right) = {\cos}^{-} 1 \left(x\right) = \theta$
So the given problem simplifies down to $\frac{7 \pi}{4}$, the original $\theta$ value.