Question

If $\tan \theta =\frac{5}{6}$ and $\tan \varphi =\frac{1}{11}$, then $\theta +\varphi$ is ___

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B ${45}^{\circ }$

We know that $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

Now, $\tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }$

$=\frac{\frac{5}{6}+\frac{1}{11}}{1-\frac{5}{6}\times \frac{1}{11}}$

$=\frac{55+6}{66-5}$

$=\frac{61}{61}$

$=1$

$\Rightarrow \tan (\theta +\varphi )=1$

We know that $\tan {45}^{\circ }=1$

$\therefore \tan (\theta +\varphi )=\tan {45}^{\circ }$

$\Rightarrow \theta +\varphi ={45}^{\circ }$