# How do you find an equivalent algebraic expression for the composition cos(arcsin(x))?

Dec 6, 2015

$\sqrt{1 - {x}^{2}}$.

#### Explanation:

Since ${\cos}^{2} \left(x\right) = 1 - {\sin}^{2} \left(x\right)$, you have that $\cos \left(x\right) = \setminus \pm \setminus \sqrt{1 - {\sin}^{2} \left(x\right)}$. So, your expression becomes

sqrt(1-sin^2(arcsin(x))

And since $\sin \left(\arcsin \left(x\right)\right) = x$, then ${\sin}^{2} \left(\arcsin \left(x\right)\right) = {x}^{2}$.

So, your expression becomes $\sqrt{1 - {x}^{2}}$.