Verify the Identity

Problem

cot(x−y)=(1+cot(y)*cot(x))/(cot(y)−cot(x))

Solution

1. Use the reciprocal identity to rewrite the cotangent function in terms of the tangent function.

cot(x−y)=1/tan(x−y)

2. Apply the tangent subtraction formula to the denominator of the expression.

tan(x−y)=(tan(x)−tan(y))/(1+tan(x)*tan(y))

3. Substitute the formula back into the reciprocal identity and simplify the fraction.

cot(x−y)=(1+tan(x)*tan(y))/(tan(x)−tan(y))

4. Convert tangent to cotangent by substituting tan(x)=1/cot(x) and tan(y)=1/cot(y) into the expression.

cot(x−y)=(1+1/cot(x)⋅1/cot(y))/(1/cot(x)−1/cot(y))

5. Multiply the numerator and denominator by the common denominator cot(x)*cot(y) to clear the complex fraction.

cot(x−y)=((1+1/(cot(x)*cot(y)))⋅cot(x)*cot(y))/((1/cot(x)−1/cot(y))⋅cot(x)*cot(y))

6. Distribute and simplify the terms to reach the final form of the identity.

cot(x−y)=(cot(x)*cot(y)+1)/(cot(y)−cot(x))

Final Answer

cot(x−y)=(1+cot(y)*cot(x))/(cot(y)−cot(x))

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