Simplify csc(arctan(u))

Problem

csc(arctan(u))

Solution

1. Define a right triangle where θ=arctan(u) which implies tan(θ)=u/1

2. Identify the sides of the triangle based on the definition of tangent as opposite/adjacent so the opposite side is u and the adjacent side is 1

3. Calculate the hypotenuse using the Pythagorean theorem: c=√(,u2+1)=√(,u2+1)

4. Apply the definition of the cosecant function, which is csc(θ)=hypotenuse/opposite

5. Substitute the side lengths into the cosecant ratio to find the simplified expression.

Final Answer

csc(arctan(u))=√(,u2+1)/u

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