For example, say

sin(\(\displaystyle \theta\)) = \(\displaystyle (1 - x^{2})^{(1/4)}\)

The corresponding value of cosine would be

cos(\(\displaystyle \theta\)) = \(\displaystyle \sqrt {(1 - ((1 - x^{2})^{(1/4)})^{2})}\) = \(\displaystyle \sqrt {(1 - (1 - x^{2})^{(1/2)})}\) = \(\displaystyle \sqrt {(1 - \sqrt{(1 - x^{2})})}\)

Or, another curve I am looking at:

sin(\(\displaystyle \theta\)) = \(\displaystyle (1 - x^{2})^{(3/4)}\)

How to reduce the corresponding expression for cosine?

Or, another one, simply that :

cos(\(\displaystyle \theta\)) = \(\displaystyle \sqrt{(1 - sin(\theta))}\)

Is there any way to simplify these expressions to eliminate the radical?

I was looking at expressing the sine functions in terms of r*exp(x\(\displaystyle \theta\)), but didn't get far with it. Perhaps I am overlooking something simple. Any advice?