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

Problem

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 expression.

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

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

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

4. Simplify the first term by canceling cos(x) and multiply the second term.

sin(x)+cos2(x)/sin(x)

5. Find a common denominator to combine the terms into a single fraction.

sin2(x)/sin(x)+cos2(x)/sin(x)

(sin2(x)+cos2(x))/sin(x)

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

1/sin(x)

7. Identify the reciprocal identity for the resulting fraction.

1/sin(x)=csc(x)

Final Answer

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

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