Simplify tan(arccos(4x))

Problem

tan(arccos(4*x))

Solution

1. Identify the inner function as an angle θ=arccos(4*x) which implies cos(θ)=4*x

2. Represent the relationship using a right triangle where the adjacent side is 4*x and the hypotenuse is 1

3. Apply the Pythagorean theorem to find the opposite side b of the triangle: b2+(4*x)2=1

4. Solve for the opposite side: b=√(,1−16*x2)

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

6. Substitute the side lengths into the ratio: tan(θ)=√(,1−16*x2)/(4*x)

Final Answer

tan(arccos(4*x))=√(,1−16*x2)/(4*x)

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