If $\int \frac{1}{\sqrt{2 a x-x^2}} d x=fog(x)+C$, then

$f(x)=\sin ^{-1} x$, and $g(x)=\frac{x-a}{a}$

We have,

$\int \frac{1}{\sqrt{2 a x-x^2}} d x=\int \frac{1}{\sqrt{a^2-(x-a)^2}} d(x-a)=\sin ^{-1}\left(\frac{x-a}{a}\right)+C$

$\Rightarrow fog(x)+C=\sin ^{-1}\left(\frac{x-a}{a}\right)+C$

$\Rightarrow f(g(x))=\sin ^{-1}\left(\frac{x-a}{a}\right)$

$\Rightarrow f(x)=\sin ^{-1} x \text { and } g(x)=\frac{x-a}{a}$