# Thread: cosine of arcsine of x?

1. ## cosine of arcsine of x?

I am doing a differential equation, where I have an initial condition, so I find the general solution, then I need to find explicit solution, and I have everything correctly, but I would like to know the rationale or logic of this part in my solution:

cos(arcsine(x))= sqrt(1-x^2)

TYVM

3. ## Re: cosine of arcsine of x?

Do you know that $sin^2(x)+ cos^2(x)= 1$?

So $cos(x)= \sqrt{1- sin^2(x)}$.

And $cos(arcsin(x))= \sqrt{1- sin^2(arcsin(x))}= \pm\sqrt{1- x^2}$.

4. ## Re: cosine of arcsine of x?

Originally Posted by HallsofIvy
Do you know that $sin^2(x)+ cos^2(x)= 1$?

So $cos(x)= \sqrt{1- sin^2(x)}$.

And $cos(arcsin(x))= \sqrt{1- sin^2(arcsin(x))}= \pm\sqrt{1- x^2}$.
Yes I know that identity, lol.

This makes perfect sense now, I totally see it. the sine and arcsine cancel leaving sqrt(1-x^2).