Firstly, note that for the range of arcsin(\frac{x}{5})arcsin(x5), the cosine function is purely positive.
Thus, I can say,
cos(arcsin(\frac{x}{5})) = sqrt{1-sin^2(arcsin(\frac{x}{5}))}cos(arcsin(x5))=√1−sin2(arcsin(x5)), and also for the domain of x, sin(arcsin(\frac{x}{5})) = \frac{x}{5}sin(arcsin(x5))=x5 since the arcsine function outputs values ranging from -\frac{pi}{2}−π2 to \frac{pi}{2}π2.
This can be verified graphically, the red one being cos(arcsin(\frac{x}{5}))cos(arcsin(x5)), green cos(\frac{x}{5})cos(x5), and blue arcsin(\frac{x}{5})arcsin(x5):
Therefore,
cos(arcsin(\frac{x}{5})) = sqrt{1-\frac{x^2}{25})cos(arcsin(x5))=√1−x225
= \frac{sqrt{25-x^2}}{5}=√25−x25.