Simplify csc(x)tan(x)+sec(-x)

Problem

csc(x)*tan(x)+sec(−x)

Solution

1. Apply trigonometric identities to rewrite the first term using sine and cosine.

csc(x)=1/sin(x)

tan(x)=sin(x)/cos(x)

2. Multiply the terms in the first part of the expression and simplify by canceling common factors.

1/sin(x)⋅sin(x)/cos(x)=1/cos(x)

3. Apply the even/odd identity for the secant function, noting that secant is an even function.

sec(−x)=sec(x)

4. Rewrite the expression using the simplified components.

1/cos(x)+sec(x)

5. Substitute the reciprocal identity for the first term to combine like terms.

1/cos(x)=sec(x)

sec(x)+sec(x)=2*sec(x)

Final Answer

csc(x)*tan(x)+sec(−x)=2*sec(x)

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