How do you simplify the expression cos(arcsin(x/5)) cos(arcsin(x5))?

1 Answer
Jul 18, 2016

cos(arcsin(\frac{x}{5})) = \frac{sqrt{25-x^2}}{5}cos(arcsin(x5))=25x25.

Explanation:

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))=1sin2(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): enter image source here

Therefore,

cos(arcsin(\frac{x}{5})) = sqrt{1-\frac{x^2}{25})cos(arcsin(x5))=1x225

= \frac{sqrt{25-x^2}}{5}=25x25.