Question

If $(1+\tan \theta )(1+\tan \varphi )=2$, then $(\theta +\varphi )$ is equal to

• A. ${30}^o$
• B. ${45}^o$
• C. ${60}^o$
• D. ${75}^o$

Answer 1

Answer: (B)

${45}^o$

We have $(1+\tan \theta )(1+\tan \varphi )=2$
$\Rightarrow 1+\tan \theta +\tan \varphi +\tan \theta \cdot \tan \varphi =2$
$\Rightarrow \tan \theta +\tan \varphi =1-\tan \theta \cdot \tan \varphi$
$\Rightarrow \frac{\tan \theta +\tan \varphi }{1-\tan \theta \cdot \tan \varphi }=1$
$\Rightarrow \tan (\theta +\varphi )=\tan \frac{\pi }{4}$
$\Rightarrow \theta +\varphi =\frac{\pi }{4}={45}^o$