Verify the Identity cot(2u)=(cot(u)^2+1)/(2cot(u))

Problem

cot(2*u)=(cot2(u)−1)/(2*cot(u))

Solution

1. Identify the standard double-angle identity for the tangent function.

tan(2*u)=(2*tan(u))/(1−tan2(u))

2. Apply the reciprocal relationship between cotangent and tangent, where cot(2*u)=1/tan(2*u)

cot(2*u)=(1−tan2(u))/(2*tan(u))

3. Substitute tan(u)=1/cot(u) into the expression to express everything in terms of cotangent.

cot(2*u)=(1−(1/cot(u))2)/(2*(1/cot(u)))

4. Simplify the numerator by finding a common denominator.

1−1/cot2(u)=(cot2(u)−1)/cot2(u)

5. Divide the simplified numerator by the denominator 2*(1/cot(u))=2/cot(u)

cot(2*u)=(cot2(u)−1)/cot2(u)⋅cot(u)/2

6. Cancel the common factor of cot(u) from the numerator and denominator to reach the final form.

cot(2*u)=(cot2(u)−1)/(2*cot(u))

Final Answer

cot(2*u)=(cot2(u)−1)/(2*cot(u))

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