# How do you simplify sin(x+y)+tan(x-y)*cos(x+y) to trigonometric functions of x and y?

Feb 5, 2016

Use the following identities:

[1] $\text{ } \tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)$

[2] $\text{ } \sin \left(x + y\right) = \sin \left(x\right) \cdot \cos \left(y\right) + \cos \left(x\right) \cdot \sin \left(y\right)$

[3] $\text{ } \sin \left(x - y\right) = \sin \left(x\right) \cdot \cos \left(y\right) - \cos \left(x\right) \cdot \sin \left(y\right)$

[4] $\text{ } \cos \left(x + y\right) = \cos \left(x\right) \cdot \cos \left(y\right) - \sin \left(x\right) \cdot \sin \left(y\right)$

[5] $\text{ } \cos \left(x - y\right) = \cos \left(x\right) \cdot \cos \left(y\right) + \sin \left(x\right) \cdot \sin \left(y\right)$

[6] $\text{ } {\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$

Thus, you can transform:

$\sin \left(x + y\right) + \tan \left(x - y\right) \cdot \cos \left(x + y\right)$

stackrel("[1] ")(=) " " sin(x+y) + sin(x-y) / cos(x-y) * cos(x+y)

$= \text{ } \sin \left(x + y\right) + \frac{\sin \left(x - y\right) \cos \left(x + y\right)}{\cos} \left(x - y\right)$

$\stackrel{\text{[2],[3],[4],[5]}}{=} \sin \left(x\right) \cos \left(y\right) + \cos \left(x\right) \sin \left(y\right) + \frac{\left(\sin \left(x\right) \cos \left(y\right) - \cos \left(x\right) \sin \left(y\right)\right) \cdot \left(\cos \left(x\right) \cos \left(y\right) - \sin \left(x\right) \sin \left(y\right)\right)}{\cos \left(x\right) \cos \left(y\right) + \sin \left(x\right) \sin \left(y\right)}$

$= \text{ } \sin \left(x\right) \cos \left(y\right) + \cos \left(x\right) \sin \left(y\right) + \frac{\sin \left(x\right) \cos \left(x\right) {\cos}^{2} \left(y\right) - \sin \left(y\right) \cos \left(y\right) {\cos}^{2} \left(x\right) - \sin \left(y\right) \cos \left(y\right) {\sin}^{2} \left(x\right) + \sin \left(x\right) \cos \left(x\right) {\sin}^{2} \left(y\right)}{\cos \left(x\right) \cos \left(y\right) + \sin \left(x\right) \sin \left(y\right)}$

$= \text{ } \sin \left(x\right) \cos \left(y\right) + \cos \left(x\right) \sin \left(y\right) + \frac{\textcolor{g r e e n}{\sin \left(x\right) \cos \left(x\right) {\cos}^{2} \left(y\right)} - \sin \left(y\right) \cos \left(y\right) {\cos}^{2} \left(x\right) - \sin \left(y\right) \cos \left(y\right) {\sin}^{2} \left(x\right) + \textcolor{g r e e n}{\sin \left(x\right) \cos \left(x\right) {\sin}^{2} \left(y\right)}}{\cos \left(x\right) \cos \left(y\right) + \sin \left(x\right) \sin \left(y\right)}$

$= \text{ } \sin \left(x\right) \cos \left(y\right) + \cos \left(x\right) \sin \left(y\right) + \frac{\sin \left(x\right) \cos \left(x\right) \left({\cos}^{2} \left(y\right) + {\sin}^{2} \left(y\right)\right) - \sin \left(y\right) \cos \left(y\right) \left({\cos}^{2} \left(x\right) + {\sin}^{2} \left(x\right)\right)}{\cos \left(x\right) \cos \left(y\right) + \sin \left(x\right) \sin \left(y\right)}$

 stackrel("[6] ")(=)" " sin(x) cos(y) + cos(x) sin(y) + (sin(x)cos(x) - sin(y)cos(y))/(cos(x)cos(y) + sin(x)sin(y))

Unfortunately, I don't see how you could efficiently simplify this expression any further.

Hope that helped!