Simplify (sec(x)+csc(x))(cos(x)-sin(x))

Problem

(sec(x)+csc(x))*(cos(x)−sin(x))

Solution

1. Expand the expression using the FOIL method (First, Outer, Inner, Last).

sec(x)*cos(x)−sec(x)*sin(x)+csc(x)*cos(x)−csc(x)*sin(x)

2. Apply reciprocal identities where sec(x)=1/cos(x) and csc(x)=1/sin(x)

1/cos(x)*cos(x)−1/cos(x)*sin(x)+1/sin(x)*cos(x)−1/sin(x)*sin(x)

3. Simplify the products of the reciprocal functions.

1−sin(x)/cos(x)+cos(x)/sin(x)−1

4. Cancel the constant terms 1 and −1

−sin(x)/cos(x)+cos(x)/sin(x)

5. Substitute quotient identities where tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x)

−tan(x)+cot(x)

6. Rearrange the terms for the final simplified form.

cot(x)−tan(x)

Final Answer

(sec(x)+csc(x))*(cos(x)−sin(x))=cot(x)−tan(x)

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