Simplify cos(sin(-1x))

Problem

cos(sin(x)(−1))

Solution

1. Identify the expression as a composition of a trigonometric function and an inverse trigonometric function, where θ=sin(x)(−1)

2. Define the relationship based on the inverse sine function, which implies sin(θ)=x for θ in the interval [−π/2,π/2]

3. Use the Pythagorean identity cos2(θ)+sin2(θ)=1 to relate the two functions.

4. Solve for cos(θ) by substituting sin(θ)=x into the identity.

cos2(θ)+x2=1

5. Isolate the cosine term.

cos2(θ)=1−x2

6. Take the square root of both sides. Since θ is in the interval [−π/2,π/2] the cosine of θ must be non-negative.

cos(θ)=√(,1−x2)

Final Answer

cos(sin(x)(−1))=√(,1−x2)

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