Simplify 2/(cot(x)-tan(x))

Problem

2/(cot(x)−tan(x))

Solution

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

2/(cos(x)/sin(x)−sin(x)/cos(x))

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

2/(cos2(x)−sin2(x))/(sin(x)*cos(x))

3. Invert and multiply to simplify the complex fraction.

(2*sin(x)*cos(x))/(cos2(x)−sin2(x))

4. Apply double-angle identities where 2*sin(x)*cos(x)=sin(2*x) and cos2(x)−sin2(x)=cos(2*x)

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

5. Simplify the resulting expression using the identity sin(θ)/cos(θ)=tan(θ)

tan(2*x)

Final Answer

2/(cot(x)−tan(x))=tan(2*x)

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