Precalculus, Chapter 5, 5.2, Section 5.2, Problem 38

Verify the identity: [tan(x)+tan(y)]/[1-tan(x)tan(y)]=[cot(x)+cot(y)]/[cot(x)cot(y)-1]
Divide every term on the left side of the equation by tan(x)tan(y)
[[tan(x)/{tan(x)tan(y)]]+[tan(y)/[tan(x)tan(y)]]]/[[1/[tan(x)tan(y)]]-[[tan(x)tan(y)]/[tan(x)tan(y)]]]=[cot(x)+cot(y)]/[cot(x)cot(y)-1]
Simplify each term.
[[1/tan(y)]+[1/tan(x)]]/[[1/[tan(x)tan(y)]]-1]=[cot(x)+cot(y)]/[cot(x)cot(y)-1]
Simplify the left side of the equation using the reciprocal identity.
[cot(y)+cot(x)]/[cot(x)cot(y)-1]=[cot(x)+cot(y)]/[cot(x)cot(y)-1]
[cot(x)+cot(y)]/[cot(x)cot(y)-1]=[cot(x)+cot(y)]/[cot(x)cot(y)-1]