Simplify cos(arcsin(1/x))

Problem

cos(arcsin(1/x))

Solution

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

cos2(θ)+(1/x)2=1

4. Isolate the cos(θ) term by subtracting the squared sine term from both sides.

cos2(θ)=1−1/(x2)

5. Simplify the right side of the equation by finding a common denominator.

cos2(θ)=(x2−1)/(x2)

6. Take the square root of both sides to find cos(θ) assuming the result is in the range where the cosine is positive (the standard range for arcsin() is [−π/2,π/2].

cos(θ)=√(,(x2−1)/(x2))

7. Simplify the radical expression by taking the square root of the denominator.

cos(θ)=√(,x2−1)/|x|

Final Answer

cos(arcsin(1/x))=√(,x2−1)/|x|

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