# Proof of cos(arcsin(x))

• Nov 14th 2010, 11:32 AM
Proof of cos(arcsin(x))
Hello, I was wondering if someone could help me with the proof of $cos(arcsin(x))=sqrt(1-x^2)$. I'm a little confused, I was using cos=sqrt(1-x^2) and arcsin=1/(sqrt(1-x^2)) and getting no where. A trigonometric proof isn't needed but if it's the only way then that's fine. Thanks for all help!
• Nov 14th 2010, 11:46 AM
skeeter
Quote:

Originally Posted by Dudealadude
Hello, I was wondering if someone could help me with the proof of $cos(arcsin(x))=sqrt(1-x^2)$. I'm a little confused, I was using cos=sqrt(1-x^2) and arcsin=1/(sqrt(1-x^2)) and getting no where. A trigonometric proof isn't needed but if it's the only way then that's fine. Thanks for all help!

let $y = \arcsin(x)$

this can also be written as ...

$\displaystyle \sin(y) = x = \frac{x}{1} = \frac{opposite \, side}{hypotenuse}$

$\displaystyle \cos[\arcsin(x)] = \cos(y) = \frac{adjacent \, side}{hypotenuse} = \frac{\sqrt{1 - x^2}}{1} = \sqrt{1 - x^2}$