Simplify tan(arccos(2x))

Problem

tan(arccos(2*x))

Solution

1. Identify the inner function as an angle θ=arccos(2*x) which implies cos(θ)=2*x

2. Represent the relationship using a right triangle where the adjacent side is 2*x and the hypotenuse is 1

3. Apply the Pythagorean theorem to find the opposite side b of the triangle: b=√(,1−(2*x)2)=√(,1−4*x2)

4. Use the definition of the tangent function, which is the ratio of the opposite side to the adjacent side: tan(θ)=opposite/adjacent

5. Substitute the side lengths into the ratio to get √(,1−4*x2)/(2*x)

Final Answer

tan(arccos(2*x))=√(,1−4*x2)/(2*x)

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