Question

If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ (where $\theta$ and $\varphi$ both belongs to $1^{\text{st}}$ quadrant), then which of the following is/are true?

• A. $\cot (\theta +\varphi )=\frac{63}{65}$
• B. $\tan (\theta +\varphi )=\frac{56}{33}$
• C. $\sin (\theta +\varphi )=\frac{56}{65}$
• D. $\cos (\theta -\varphi )=\frac{63}{65}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\tan (\theta +\varphi )=\frac{56}{33}$
C $\sin (\theta +\varphi )=\frac{56}{65}$
D $\cos (\theta -\varphi )=\frac{63}{65}$

$\sin \theta =\frac{3}{5}\Rightarrow \cos \theta =\sqrt{1-{\sin }^2\theta }=\frac{4}{5}\Rightarrow \tan \theta =\frac{3}{4}$
Similarily
$\cos \varphi =\frac{12}{13},\Rightarrow \sin \varphi =\frac{5}{13},\tan \varphi =\frac{5}{12}$
$\tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }=\frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}\times \frac{5}{12}}=\frac{56}{33}$
$\sin (\theta +\varphi )=\sin \theta \cos \varphi +\cos \theta \sin \varphi =\frac{56}{65}$
$\cos (\theta -\varphi )=\cos \theta \cos \varphi +\sin \theta \sin \varphi =\frac{63}{65}$