Simplify tan(arccos(5x))

Problem

tan(arccos(5*x))

Solution

1. Identify the inner function as an angle θ=arccos(5*x) which implies cos(θ)=5*x

2. Represent the relationship using a right triangle where the adjacent side is 5*x and the hypotenuse is 1

3. Determine the opposite side of the triangle using the Pythagorean theorem: a^2+b^2=c^2

4. Solve for the opposite side: opposite=√(,1^2−(5*x)^2)=√(,1−25*x^2)

5. Apply 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 expressions for the sides to find the final simplified form.

Final Answer

tan(arccos(5*x))=√(,1−25*x^2)/(5*x)

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