Simplify tan(arccos(x))

Problem

tan(arccos(x))

Solution

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

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

3. Apply the Pythagorean theorem to find the opposite side length: a^2+b^2=c^2⇒opposite^2+x^2=1^2

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

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−x^2)/x

Final Answer

tan(arccos(x))=√(,1−x^2)/x

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