Answer
$\dfrac{3 \pi}{4}$
Work Step by Step
Let us substitute:
$g^{-1}(f (\dfrac{ 7 \pi}{4})=\cos (\sin^{-1}(\dfrac{ 7 \pi}{4}))$
From the unit circle, we know that:
$\sin \dfrac{ 7 \pi}{4}= (\dfrac{-\sqrt 2}{2})$
We know that the trigonometric function $\cos^{-1}$ has the domain $[-1,1]$ and the range $[0, \pi]$.
Therefore, $\cos^{-1} (\sin \dfrac{7 \pi}{4})=\cos^{-1} (\dfrac{-\sqrt 2}{2}) =\dfrac{3 \pi}{4}$ because $\cos \dfrac{3 \pi}{4}=\dfrac{-\sqrt 2}{2}$ and $\dfrac{3 \pi}{4}$ lies in the range of $\cos^{-1} x$.
