Question

If $\theta +\varphi =\frac{\pi }{4}$, then $\left(1+\tan \theta \right)\left(1+\tan \varphi \right)$ is equal to

• A. 1
• B. 2
• C. 5/2
• D. 1/3

Answer 1

Answer: (B)

$(1+\tan \theta )(1+\tan \varphi )$
$=(1+\tan \theta )\left[1+\tan \left(\frac{\pi }{4}-\theta \right)\right]$
$=(1+\tan \theta )\left[1+\frac{1-\tan \theta }{1+\tan \theta }\right]$
$=(1+\tan \theta )\cdot \frac{2}{(1+\tan \theta )}=2$