Simplify tan(x)(cos(x)-csc(x))

Problem

tan(x)*(cos(x)−csc(x))

Solution

1. Distribute the tan(x) term to both terms inside the parentheses.

tan(x)*cos(x)−tan(x)*csc(x)

2. Rewrite the trigonometric functions in terms of sin(x) and cos(x) using the identities tan(x)=sin(x)/cos(x) and csc(x)=1/sin(x)

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

3. Simplify each term by canceling common factors in the numerators and denominators.

sin(x)−1/cos(x)

4. Apply the reciprocal identity sec(x)=1/cos(x) to the second term.

sin(x)−sec(x)

Final Answer

tan(x)*(cos(x)−csc(x))=sin(x)−sec(x)

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