Simplify cos(arcsin(2x))

Problem

cos(arcsin(2*x))

Solution

1. Identify the inner function and set it equal to an angle θ Let θ=arcsin(2*x)

2. Rewrite the inverse trigonometric expression as a trigonometric equation. This gives sin(θ)=2*x which can be viewed as sin(θ)=(2*x)/1

3. Relate the sine ratio to a right triangle where the opposite side is 2*x and the hypotenuse is 1

4. Apply the Pythagorean theorem to find the adjacent side a

a2+(2*x)2=1

a2+4*x2=1

a=√(,1−4*x2)

5. Evaluate the original expression cos(θ) using the ratio of the adjacent side over the hypotenuse.

cos(θ)=√(,1−4*x2)/1

cos(θ)=√(,1−4*x2)

Final Answer

cos(arcsin(2*x))=√(,1−4*x2)

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