Simplify 1/(tan(x)+cot(x))

Problem

1/(tan(x)+cot(x))

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x)

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

2. Find a common denominator for the terms in the denominator of the main fraction.

1/(sin2(x)+cos2(x))/(sin(x)*cos(x))

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

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

4. Simplify the complex fraction by taking the reciprocal.

sin(x)*cos(x)

5. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) which implies sin(x)*cos(x)=1/2*sin(2*x)

1/2*sin(2*x)

Final Answer

1/(tan(x)+cot(x))=1/2*sin(2*x)

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