Question

Question in pictureA) 16/63B)-16/63C) 63/16D) -63/16

Answer 1

ANSWER
$\tan(x + y) = - \frac{63}{16}$


EXPLANATION


We were given that,

$\csc(x) = \frac{5}{3}$

This implies that,

$\sin(x) = \frac{3}{5}$

We use the Pythagorean identity

$\sin^{2} (x) + \cos^{2} (x)= 1$
to get,


$\cos(x) = \sqrt{1 - ( { \frac{3}{5} })^{2}} = \frac{4}{5}$


We were also given that,


$\cos(y) = \frac{5}{13}$

This means that,


$\sin(y) = \sqrt{1 - {( \frac{5}{13}) }^{2} } = \frac{12}{13}$

This is because,


$0 < \: x \: < \frac{\pi}{2}$


$0 < \: y \: < \frac{\pi}{2}$

This angles are in the first quadrant so we pick the positive values.

$\tan(x + y) = \frac{ \sin(x + y) }{ \cos(x + y) }$


$\tan(x + y) = \frac{ \sin(x ) \cos(y) + \sin(y) \cos(x) }{ \cos(x) \cos(y) - \sin(x) \sin(y) }$



$\tan(x + y) = \frac{ \frac{3}{5} \times \frac{5}{13} + \frac{12}{13} \times \frac{4}{5} }{ \frac{4}{5} \times \frac{5}{13} - \frac{3}{5} \times \frac{12}{13} }$



$\tan(x + y) = - \frac{63}{16}$

The correct answer is D