Question

If $A+B=\frac{\pi }{4}$, then $(1+\tan A)(1+\tan B)=$

• A. $1$
• B. $2$
• C. $\infty$
• D. $-2$

Answer 1

The correct option is B

$2$

Explanation for the correct option:

Find the value of $(1+\tan A)(1+\tan B)$:

Given, $A+B=\frac{\pi }{4}$

Take “tan” on both sides,

$\tan (A+B)=\tan \frac{\pi }{4}$

$\Rightarrow$ $\frac{t\mathrm{an}A+\tan B}{1–\tan A\tan B}=1$

$\Rightarrow$ $\tan A+\tan B=1–\tan A\tan B$

$\Rightarrow$$\tan A+\tan B+\tan A\tan B=1$ ….(i)

$\begin{array}{rcl}\therefore (1+\tan A)(1+\tan B)& =& 1+\tan A+\tan B+\tan A\tan B\\ & =& 1+1\\ & =& \mathbf{2}\end{array}$ [From equation(1)]

Hence, Option ‘B’ is Correct.