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The tangent addition formula states that tan(x+y) = (tan(x) + tan(y))/(1 - tan(x)tan(y)).

To prove the tangent addition formula, we start with the identity:

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

Dividing both sides by cos(x)cos(y), we get:

tan(x+y) = (sin(x+y))/(cos(x)cos(y))

= (sin(x)cos(y) + cos(x)sin(y))/(cos(x)cos(y))

= (sin(x)/cos(x) + sin(y)/cos(y))/(1 - (sin(x)sin(y))/(cos(x)cos(y)))

= (tan(x) + tan(y))/(1 - tan(x)tan(y))

Therefore, we have proven the tangent addition formula.

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