Simplify cos(arcsin(7/x))

Problem

cos(arcsin(7/x))

Solution

1. Define a variable θ such that θ=arcsin(7/x) This implies that sin(θ)=7/x

2. Identify the goal as finding cos(θ) based on the trigonometric identity sin2(θ)+cos2(θ)=1

3. Rearrange the identity to solve for cos(θ) which gives cos(θ)=√(,1−sin2(θ)) Note that the range of arcsin() is [−π/2,π/2] where cosine is non-negative.

4. Substitute the value of sin(θ)=7/x into the expression.

cos(θ)=√(,1−(7/x)2)

5. Simplify the expression by squaring the fraction and finding a common denominator.

cos(θ)=√(,1−49/(x2))

cos(θ)=√(,(x2−49)/(x2))

6. Extract the square root from the denominator.

cos(θ)=√(,x2−49)/|x|

Final Answer

cos(arcsin(7/x))=√(,x2−49)/|x|

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