If (1 + tanA)(1 + tanB) = 2, then what will be the value of tan(A + B)?

Correct option is D

Given:

​$$(1 + \tan A)(1 + \tan B) = 2$$  

To find: The value of  $$\tan(A + B)$$​​

Formula Used:

​$$\tan(A + B) = \frac{\tan A + \tan B}{1 - (\tan A)(\tan B)}$$​​

Solution:

​$$\begin{aligned}& (1 + \tan A)(1 + \tan B) = 2 \\& \implies 1 + \tan A + \tan B + \tan A \tan B = 2 \\& \implies \tan A + \tan B = 1 - \tan A \tan B \\& \implies \left[\frac{\tan A + \tan B}{1 - \tan A \tan B}\right] = 1 \\& \implies \tan(A + B) = 1 \\& \therefore \text{The value of } \tan(A + B) = 1.\end{aligned}$$

Alternate Solution:

Put A = $$45^\circ$$  and B= $$0^\circ$$ in the given equation:

we get,

(1 + 1) (1 + 0) = 2  (Satisfied)

​$$\tan (A + B) = \tan (45^\circ + 0^\circ) = \tan 45^\circ = 1\\\tan (A + B) = 1$$

Hence, option (d) is the correct answer.