Question

If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ where $\theta ,\varphi \in (0,\pi /2)$, then the value of $\tan (\theta +\varphi )$ is

• A. $\frac{8}{33}$
• B. $\frac{7}{33}$
• C. $\frac{33}{56}$
• D. $\frac{56}{33}$

Answer 1

The correct option is D $\frac{56}{33}$

We have $\theta ,\varphi \in (0,\pi /2)$
So $\sin \theta =\frac{3}{5}\Rightarrow \cos \theta =\sqrt{1-{\sin }^2\theta }=\frac{4}{5}\Rightarrow \tan \theta =\frac{3}{4}$
Similarily,
$\cos \varphi =\frac{12}{13},\Rightarrow \sin \varphi =\frac{5}{13},\tan \varphi =\frac{5}{12}$
$\tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }=\frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}\times \frac{5}{12}}=\frac{56}{33}$