Determine if Odd, Even, or Neither y=csc(x)

Problem

y=csc(x)

Solution

1. Recall the definition of even and odd functions. A function ƒ(x) is even if ƒ*(−x)=ƒ(x) and odd if ƒ*(−x)=−ƒ(x)

2. Substitute −x for x in the given function.

ƒ*(−x)=csc(−x)

3. Use the reciprocal identity to rewrite the cosecant function in terms of the sine function.

csc(−x)=1/sin(−x)

4. Apply the odd function identity for sine, which states that sin(−x)=−sin(x)

1/sin(−x)=1/(−sin(x))

5. Simplify the expression to compare it to the original function.

1/(−sin(x))=−1/sin(x)

6. Substitute the cosecant identity back into the expression.

−1/sin(x)=−csc(x)

7. Conclude that since ƒ*(−x)=−ƒ(x) the function is odd.

Final Answer

csc(−x)=−csc(x)⇒Odd

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