If $\frac{1}{1+x}+\frac{2}{2+y}+\frac{1009}{1009+z}=1$, then the value of $\frac{x}{1+x}+\frac{y}{2+y}+\frac{z}{1009+z}$ is equal to:

We notice that each term in the required expression can be rewritten by subtracting the corresponding term of the given equation from $1$:
$$\frac{x}{1+x}=1-\frac{1}{1+x}$$

$$\frac{y}{2+y}=1-\frac{2}{2+y}$$

$$\frac{z}{1009+z}=1-\frac{1009}{1009+z}$$

Summing these individual equations up:

$$\frac{x}{1+x}+\frac{y}{2+y}+\frac{z}{1009+z}=\left(1-\frac{1}{1+x}\right)+\left(1-\frac{2}{2+y}\right)+\left(1-\frac{1009}{1009+z}\right)$$

$$=(1+1+1)-\left(\frac{1}{1+x}+\frac{2}{2+y}+\frac{1009}{1009+z}\right)$$

Substitute the value from the given equation, which is equal to $1$:

$$=3-1=2$$