Answer
$\dfrac{12}{13}$
Work Step by Step
We know that
$g(x)=\cos(x), f(x)=\sin(x)$
Thus we can write:
$g(f^{-1} (\dfrac{5}{13}))=\sin (\cos^{-1}(\dfrac{5}{13}))$
Suppose that $\theta=\cos^{-1} (\dfrac{5}{13})$
We know from SOH-CAH-TOA:
$\cos{\theta}=\dfrac{adjacent}{hypotenuse}=\dfrac{5}{13}$
To find the missing opposite side, we use the Pythagorean Theorem:
$opposite^2+5^2=13^2$
$opposite=\sqrt{(13)^2-(5)^2}=\sqrt {144}=12$
Therefore,
$\sin{\theta}=\dfrac{opposite}{hypotenuse}=\dfrac{12}{13}$
