Thu gọn sin^2 alpha + sin^2 bêta + 2sin alpha sin bêta . cos (alpha + bêta) ta được:

* Hướng dẫn giải

$A={\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta .\cos \left(\alpha +\beta \right)$

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

$={\sin }^2\alpha +{\sin }^2\beta -2{\sin }^2\alpha {\sin }^2\beta +2\sin \alpha \sin \beta \cos \alpha .\cos \beta$

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

$={\sin }^2\alpha {\cos }^2\beta +{\sin }^2\beta {\cos }^2\alpha +2\sin \alpha \sin \beta \cos \alpha .\cos \beta$

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

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