Use the double–angle identity to transform to .

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 .

Add and .

Add and .

Apply the distributive property.

Multiply by .

Multiply .

Multiply by .

Multiply by .

Add and .

Expand cos(2arccos(x))