# Expand cos(2arccos(x)) Use the doubleangle identity to transform to .
Simplify each term.
The functions cosine and arccosine are inverses.
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Divide by .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply .
Multiply by .
Multiply by .     