Simplify cot(2x)

Problem

cot(2*x)

Solution

1. Identify the reciprocal identity for the cotangent function.

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

2. Apply the double-angle formula for the tangent function, which states tan(2*x)=(2*tan(x))/(1−tan2(x))

cot(2*x)=1/(2*tan(x))/(1−tan2(x))

3. Simplify the complex fraction by taking the reciprocal of the denominator.

cot(2*x)=(1−tan2(x))/(2*tan(x))

4. Rewrite the expression in terms of cotangent by dividing each term in the numerator by the denominator.

cot(2*x)=1/(2*tan(x))−tan2(x)/(2*tan(x))

5. Substitute 1/tan(x)=cot(x) and simplify the second term.

cot(2*x)=cot(x)/2−tan(x)/2

6. Combine the terms into a single fraction.

cot(2*x)=(cot(x)−tan(x))/2

7. Alternative form using the cotangent double-angle identity directly.

cot(2*x)=(cot2(x)−1)/(2*cot(x))

Final Answer

cot(2*x)=(cot2(x)−1)/(2*cot(x))

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