Simplify cos(arccos(x))

Problem

cos(arccos(x))

Solution

1. Identify the domain of the inner function. The function arccos(x) is defined for x in the interval [−1,1]

2. Apply the definition of inverse trigonometric functions. By definition, y=arccos(x) is the angle such that cos(y)=x where 0≤y≤π

3. Substitute the relationship back into the expression. Since cos(arccos(x)) asks for the cosine of the angle whose cosine is x the operations cancel each other out.

4. Simplify the expression based on the identity cos(arccos(x))=x for all x in the domain [−1,1]

Final Answer

cos(arccos(x))=x

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