Simplify sin(2arccos(x))

Problem

sin(2*arccos(x))

Solution

1. Identify the double angle identity for sine, which states sin(2*θ)=2*sin(θ)*cos(θ)

2. Substitute θ=arccos(x) into the identity to rewrite the expression.

sin(2*arccos(x))=2*sin(arccos(x))*cos(arccos(x))

3. Simplify the term cos(arccos(x)) using the property of inverse functions, which yields x

4. Evaluate the term sin(arccos(x)) by using the Pythagorean identity sin2(θ)+cos2(θ)=1

5. Solve for sin(θ) given cos(θ)=x which results in sin(θ)=√(,1−x2)

6. Combine the simplified parts into the final expression.

2*(√(,1−x2))*(x)

Final Answer

sin(2*arccos(x))=2*x√(,1−x2)

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