Tính sin alpha + sin bêta cos (alpha + bêta)/cos alpha - sin bêta sin (alpha + bêta)

* Hướng dẫn giải

Ta có: $\frac{\sin \alpha +\sin \beta \cos \left(\alpha +\beta \right)}{\cos \alpha -\sin \beta \sin \left(\alpha +\beta \right)}$

$=\frac{\sin \alpha +\frac{1}{2}\left[\sin \left(\alpha +2\beta \right)+\sin \left(-\alpha \right)\right]}{\cos \alpha +\frac{1}{2}\left[\cos \left(\alpha +2\beta \right)-\cos \left(-\alpha \right)\right]}$

$=\frac{\sin \alpha +\frac{1}{2}\left[\sin \left(\alpha +2\beta \right)-\sin \left(\alpha \right)\right]}{\cos \alpha +\frac{1}{2}\left[\cos \left(\alpha +2\beta \right)-\cos \left(\alpha \right)\right]}$

$=\frac{\sin \alpha +\frac{1}{2}\sin \left(\alpha +2\beta \right)-\frac{1}{2}\sin \alpha }{\cos \alpha +\frac{1}{2}\cos \left(\alpha +2\beta \right)-\frac{1}{2}\cos \alpha }$

$=\frac{\frac{1}{2}\sin \alpha +\frac{1}{2}\sin \left(\alpha +2\beta \right)}{\frac{1}{2}\cos \alpha +\frac{1}{2}\cos \left(\alpha +2\beta \right)}$

$=\frac{\sin \left(\alpha +2\beta \right)+\sin \alpha }{\cos \left(\alpha +2\beta \right)+\cos \alpha }$

$=\frac{2\sin \frac{\left(\alpha +2\beta +\alpha \right)}{2}\cos \frac{\left(\alpha +2\beta -\alpha \right)}{2}}{2\cos \frac{\left(\alpha +2\beta +\alpha \right)}{2}\cos \frac{\left(\alpha +2\beta -\alpha \right)}{2}}$

$=\frac{2\sin \left(\alpha +\beta \right)\cos \beta }{2\cos \left(\alpha +\beta \right)\cos \beta }$

$=\tan \left(\alpha +\beta \right)$