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

Problem

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

Solution

1. Rewrite the denominator using sine and cosine identities.

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

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

2. Find a common denominator for the terms in the denominator.

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

cot(x)+tan(x)=(cos2(x)+sin2(x))/(sin(x)*cos(x))

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

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

4. Substitute the simplified denominator and the identity csc(x)=1/sin(x) back into the original expression.

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

5. Simplify the complex fraction by multiplying by the reciprocal.

1/sin(x)⋅(sin(x)*cos(x))/1

cos(x)

Final Answer

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

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