Answer
$\frac{5}{13}$.
Work Step by Step
$g(f(\frac{12}{13}))=\cos(\sin^{-1}{\frac{12}{13}})$.
Let $\theta=\sin^{-1}{\frac{12}{13}}$, hence we know that because of the definition of the sine function:
$\sin{\theta}=\frac{12}{13}=\frac{\text{opposite}}{\text{hypotenuse}}$
The Pythagorean Theorem says that for a right triangle (if $z$ is the hypotenuse and $x,y$ are the other sides): $x^2+y^2=z^2$. Hence here: $\text{adjacent}=\sqrt{13^2-12^2}=\sqrt{25}=5.$
$\cos{\theta}=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{5}{13}$.