# Expand cos(2arctan(x)) Use the doubleangle identity to transform to .
Simplify terms.
Simplify each term.
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 .
One to any power is one.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
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.
Apply the product rule to .
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.
Cancel the common factor of and .
Multiply by .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
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 .
One to any power is one.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
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.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
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.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine the numerators over the common denominator.
Simplify the numerator.
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Expand cos(2arctan(x))     