If A+B = 45°, then the value of 2(1+ tanA)(1+ tanB) is:

The given equation is A + B = 45°.

We know that tan(45°) = 1.

So, tanA = tan(45° - B) = tan(45° - A) = 1 - tanB / 1 + tanA*tanB = 1 - tanB / 1 + tanA.

Now, substitute tanA = 1 - tanB / 1 + tanA in the given expression 2(1 + tanA)(1 + tanB).

This simplifies to 2(1 + (1 - tanB / 1 + tanA))(1 + tanB) = 2(2 - tanB / 1 + tanA).

Since tanA = 1 - tanB / 1 + tanA, we can simplify the above expression to 2(2 - tanB / 1 + (1 - tanB / 1 + tanA)).

This simplifies to 2(2 - tanB / 2 - tanB^2 / 1 + tanA) = 2(2 - tanB / 2 - tanB^2).

Since tanB = 1 - tanA, we can substitute this into the above expression to get 2(2 - (1 - tanA) / 2 - (1 - tanA)^2) = 2(2 - 1 / 2 + tanA - 1 + 2tanA - tanA^2).

This simplifies to 2(1 + tanA - 1 + 2tanA - tanA^2) = 2(2tanA - tanA^2).

Finally, this simplifies to 2(2tanA - tanA^2) = 4tanA - 2tanA^2.

So, the value of 2(1 + tanA)(1 + tanB) is 4tanA - 2tanA^2.