Simplify tan(-x)csc(x)

Problem

tan(−x)*csc(x)

Solution

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

tan(−x)*csc(x)=−tan(x)*csc(x)

2. Rewrite the functions in terms of sine and cosine using the quotient identity tan(x)=sin(x)/cos(x) and the reciprocal identity csc(x)=1/sin(x)

−tan(x)*csc(x)=−sin(x)/cos(x)⋅1/sin(x)

3. Cancel the common factor of sin(x) from the numerator and the denominator.

−sin(x)/(cos(x)*sin(x))=−1/cos(x)

4. Apply the reciprocal identity for secant, where 1/cos(x)=sec(x) to reach the simplest form.

−1/cos(x)=−sec(x)

Final Answer

tan(−x)*csc(x)=−sec(x)

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