Final answer:
To express the given expression cos(2arcsin(x)) as an algebraic expression in terms of x, we can use the identity cos^2(theta) + sin^2(theta) = 1. By substituting 2arcsin(x) for theta, we can rewrite the expression as cos^2(arcsin(x)) - sin^2(arcsin(x)).
Step-by-step explanation:
The given expression, cos(2arcsin(x)), can be rewritten using the identity cos^2(theta) + sin^2(theta) = 1. Let's start by finding the value of 2arcsin(x):
Let theta = arcsin(x). Then, sin(theta) = x. Multiplying both sides by 2, we get 2sin(theta) = 2x. Replacing sin(theta) with x, we have 2x = 2x.
Now, using the identity cos^2(theta) + sin^2(theta) = 1, we can rewrite cos(2arcsin(x)) as:
cos(2arcsin(x)) = cos^2(arcsin(x)) - sin^2(arcsin(x))