# If x and y are acute angles and sin(2x-20^@)=cos(2y+20^@), what is tan(x+y)?

Dec 20, 2017

$\tan \left(x + y\right) = 1$

#### Explanation:

$\textcolor{b l u e}{A s \textcolor{w h i t e}{.} \sin \left({90}^{\circ} - \theta\right) = \cos \theta}$

$\sin \left(2 x - {20}^{\circ}\right) = \cos \left(2 y + {20}^{\circ}\right)$ can be written as

or $\sin \left(2 x - {20}^{\circ}\right) = \sin \left({90}^{\circ} - 2 y - {20}^{\circ}\right)$

or $\sin \left(2 x - {20}^{\circ}\right) = \sin \left({70}^{\circ} - 2 y\right)$

and as $x , y \text{ and } \left(x + y\right)$ are acute angles

or $2 x - {20}^{\circ} = {70}^{\circ} - 2 y$

or $2 x + 2 y = {90}^{\circ}$

and so $x + y = {45}^{\circ}$

Hence, $\tan \left(x + y\right) = \tan {45}^{\circ} = \tan {45}^{\circ} = 1 \textcolor{w h i t e}{x} \textcolor{b r o w n}{A n s .}$