Simplify cos(arcsin(8/x))

Problem

cos(arcsin(8/x))

Solution

1. Identify the inner expression as an angle θ=arcsin(8/x) which implies sin(θ)=8/x

2. Relate the trigonometric functions using the Pythagorean identity cos2(θ)+sin2(θ)=1

3. Substitute the value of sin(θ) into the identity to get cos2(θ)+(8/x)2=1

4. Solve for cos(θ) by subtracting the squared term from both sides: cos2(θ)=1−64/(x2)

5. Simplify the expression under a common denominator: cos2(θ)=(x2−64)/(x2)

6. Take the square root to find cos(θ) assuming the range of the arcsine function (−π/2≤θ≤π/2 where the cosine is non-negative.

Final Answer

cos(arcsin(8/x))=√(,x2−64)/x

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