Simplify cos(arcsin(1/7x))

Problem

cos(arcsin(1/(7*x)))

Solution

1. Identify the inner expression as an angle θ=arcsin(1/(7*x)) which implies sin(θ)=1/(7*x)

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

3. Solve for cos(θ) by rearranging the identity to cos(θ)=√(,1−sin2(θ))

4. Substitute the value of sin(θ) into the equation:

cos(θ)=√(,1−(1/(7*x))2)

5. Simplify the expression inside the square root:

cos(θ)=√(,1−1/(49*x2))

6. Combine the terms under a common denominator:

cos(θ)=√(,(49*x2−1)/(49*x2))

7. Extract the square root of the denominator:

cos(θ)=√(,49*x2−1)/(7*|x|)

Final Answer

cos(arcsin(1/(7*x)))=√(,49*x2−1)/(7*x)

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