Simplify cos(arctan(x))

Problem

cos(arctan(x))

Solution

1. Define a variable for the inner function by letting θ=arctan(x)

2. Rewrite the relationship using the definition of the inverse tangent function, which implies tan(θ)=x for −π/2<θ<π/2

3. Construct a right triangle where the angle is θ Since tan(θ)=opposite/adjacent=x/1 let the opposite side be x and the adjacent side be 1

4. Calculate the hypotenuse using the Pythagorean theorem: a2+b2=c2 which gives 1+x2=c2 so the hypotenuse is √(,1+x2)

5. Evaluate the cosine of the angle using the ratio cos(θ)=adjacent/hypotenuse

6. Substitute the side lengths into the ratio to get cos(θ)=1/√(,1+x2)

Final Answer

cos(arctan(x))=1/√(,1+x2)

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