Simplify cos(2arctan(x))

Problem

cos(2*arctan(x))

Solution

1. Identify the expression as a double angle identity for cosine, where θ=arctan(x)

2. Define the relationship between x and θ such that tan(θ)=x

3. Apply the formula for the double angle of cosine in terms of tangent: cos(2*θ)=(1−tan2(θ))/(1+tan2(θ))

4. Substitute tan(θ)=x into the identity to express the result in terms of x

Final Answer

cos(2*arctan(x))=(1−x2)/(1+x2)

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