Simplify (csc(-x))/(1+tan(x)^2)

Problem

csc(−x)/(1+tan2(x))

Solution

1. Apply the odd-even identity for the cosecant function, which states that csc(−x)=−csc(x)

2. Apply the Pythagorean identity for the denominator, where 1+tan2(x)=sec2(x)

3. Rewrite the expression using the results from the previous steps.

(−csc(x))/sec2(x)

4. Express in terms of sine and cosine by using the reciprocal identities csc(x)=1/sin(x) and sec(x)=1/cos(x)

(−1/sin(x))/1/cos2(x)

5. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

−1/sin(x)⋅cos2(x)

6. Combine the terms into a single trigonometric expression.

−cos2(x)/sin(x)

Final Answer

csc(−x)/(1+tan2(x))=−cos2(x)/sin(x)

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