Verify the Identity cos(x)(tan(x)+cot(x))=csc(x)

Problem

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

Solution

1. Express in terms of sine and cosine by using the quotient identities tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x)

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

2. Find a common denominator for the terms inside the parentheses, which is sin(x)*cos(x)

cos(x)*((sin^2(x)+cos^2(x))/(sin(x)*cos(x)))

3. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the numerator.

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

4. Simplify the expression by canceling the cos(x) term in the numerator and denominator.

1/sin(x)

5. Apply the reciprocal identity 1/sin(x)=csc(x) to reach the final form.

csc(x)

Final Answer

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

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