`(tan(x) + tan(y))/(1 - tan(x)tan(y)) = (cot(x) + cot(y))/(cot(x)cot(y) - 1)` Verfiy the identity.

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Chapter 5, 5.2 - Problem 38 - Precalculus (3rd Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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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]`

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