Step 1: Approach
Test with a specific pair of angles that satisfy the condition, then confirm in general.
Step 2: Special case
Take $x=0$, $y=\pi/4$. Then $(1+0)(1+1)=2$.
Take $x=y=\pi/8$. Then $\tan\frac{\pi}{8}=\sqrt2-1$, so $(1+\sqrt2-1)^2=(\sqrt2)^2=2$.
Step 3: General argument
Because the product reduces to $2+(\tan x+\tan y+\tan x\tan y-1)$ and the condition $\tan(x+y)=1$ forces $\tan x+\tan y+\tan x\tan y=1$, the bracket vanishes. So the value is always 2, option (B).
Final Answer:
The product is always 2 when x + y = pi/4, option (B).
\[ \boxed{2} \]