Simplify sin(arccsc(u))

Problem

sin(arccsc(u))

Solution

1. Identify the relationship between the trigonometric function and its inverse. The function arccsc(u) represents an angle θ such that csc(θ)=u

2. Recall the reciprocal identity for the cosecant function. By definition, csc(θ)=1/sin(θ)

3. Substitute the relationship into the equation. If csc(θ)=u then 1/sin(θ)=u

4. Solve for the sine function by taking the reciprocal of both sides. This yields sin(θ)=1/u

5. Conclude that since θ=arccsc(u) the expression sin(arccsc(u)) simplifies directly to the reciprocal of the argument.

Final Answer

sin(arccsc(u))=1/u

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