Verify the Identity tan(2x)=2/(cot(x)-tan(x))

Problem

tan(2*x)=2/(cot(x)−tan(x))

Solution

1. Identify the double angle identity for the tangent function on the left side of the equation.

tan(2*x)=(2*tan(x))/(1−tan2(x))

2. Rewrite the right side of the identity by expressing the denominator in terms of tangent using the reciprocal identity cot(x)=1/tan(x)

2/(1/tan(x)−tan(x))

3. Simplify the denominator by finding a common denominator for the terms 1/tan(x) and tan(x)

1/tan(x)−tan2(x)/tan(x)=(1−tan2(x))/tan(x)

4. Divide the numerator by the simplified denominator, which is equivalent to multiplying by the reciprocal.

2⋅tan(x)/(1−tan2(x))

5. Combine the terms to see that the right side matches the double angle identity derived in step 1.

(2*tan(x))/(1−tan2(x))

Final Answer

tan(2*x)=2/(cot(x)−tan(x))

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