Simplify sin(x)cos(x)(tan(x)+cot(x))

Problem

sin(x)*cos(x)*(tan(x)+cot(x))

Solution

1. Rewrite the trigonometric functions tan(x) and cot(x) in terms of sin(x) and cos(x)

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

cot(x)=cos(x)/sin(x)

2. Substitute these expressions back into the original parentheses.

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

3. Distribute the term sin(x)*cos(x) to both terms inside the parentheses.

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

4. Cancel the common factors in the numerator and denominator for each term.

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

sin2(x)+cos2(x)

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1

1

Final Answer

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

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