Simplify csc(arcsin(x))

Problem

csc(arcsin(x))

Solution

1. Identify the relationship between the trigonometric function and its inverse. The expression involves the cosecant function, csc(θ) and the inverse sine function, arcsin(x)

2. Recall the definition of the cosecant function in terms of the sine function. The cosecant of an angle is the reciprocal of the sine of that angle.

csc(θ)=1/sin(θ)

3. Substitute the inner expression arcsin(x) for θ into the reciprocal identity.

csc(arcsin(x))=1/sin(arcsin(x))

4. Apply the property of inverse functions. For any x in the domain of the inverse sine function, [−1,1] the composition of the function and its inverse results in the original value.

sin(arcsin(x))=x

5. Simplify the expression by replacing the denominator with x

1/sin(arcsin(x))=1/x

Final Answer

csc(arcsin(x))=1/x

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