Simplify tan(x)(cot(x)+csc(x))

Problem

tan(x)*(cot(x)+csc(x))

Solution

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

tan(x)*cot(x)+tan(x)*csc(x)

2. Apply reciprocal identities for cot(x) and csc(x) and the quotient identity for tan(x)

tan(x)1/tan(x)+sin(x)/cos(x)1/sin(x)

3. Simplify the first term by canceling tan(x) and the second term by canceling sin(x)

1+1/cos(x)

4. Apply the reciprocal identity for 1/cos(x) to reach the simplest form.

1+sec(x)

Final Answer

tan(x)*(cot(x)+csc(x))=1+sec(x)

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