Simplify tan(arccos(8x))

Problem

tan(arccos(8*x))

Solution

1. Identify the inner function as an angle θ=arccos(8*x) which implies cos(θ)=8*x

2. Construct a right triangle where the adjacent side is 8*x and the hypotenuse is 1

3. Apply the Pythagorean theorem to find the opposite side b b2+(8*x)2=1

4. Solve for the opposite side: b=√(,1−(8*x)2)=√(,1−64*x2)

5. Evaluate the tangent of the angle θ using the ratio of the opposite side over the adjacent side: tan(θ)=opposite/adjacent

6. Substitute the side lengths into the ratio: tan(θ)=√(,1−64*x2)/(8*x)

Final Answer

tan(arccos(8*x))=√(,1−64*x2)/(8*x)

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