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

Problem

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

Solution

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

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

2. Apply the reciprocal identity for cosecant, which is csc(x)=1/sin(x)

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

3. Simplify the first term by canceling sin(x) and rewrite the second term as a single fraction.

1+cos(x)/sin(x)

4. Apply the cotangent identity for the ratio cos(x)/sin(x)=cot(x)

1+cot(x)

Final Answer

csc(x)*(sin(x)+cos(x))=1+cot(x)

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