The problem requires us to prove the following trigonometric identity: $tan(x+y) - tan(x) - tan(y) = tan(x)tan(y)tan(x+y)$

1. Problem Description

The problem requires us to prove the following trigonometric identity:

$tan(x+y) - tan(x) - tan(y) = tan(x)tan(y)tan(x+y)$

2. Solution Steps

We start with the tangent addition formula:

$tan(x+y) = \frac{tan(x) + tan(y)}{1 - tan(x)tan(y)}$

Multiply both sides by $1 - tan(x)tan(y)$:

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

Expand the left side:

$tan(x+y) - tan(x+y)tan(x)tan(y) = tan(x) + tan(y)$

Rearrange the terms to isolate $tan(x+y) - tan(x) - tan(y)$ on the left-hand side:

$tan(x+y) - tan(x) - tan(y) = tan(x+y)tan(x)tan(y)$

Thus, we have shown the identity.

3. Final Answer

$tan(x+y) - tan(x) - tan(y) = tan(x)tan(y)tan(x+y)$