1.恒等变换（一）

$1.\mathrm{利}\mathrm{用}1\pm \cos \theta$
$\mathrm{原}\mathrm{式}={\displaystyle \frac{2{\sin }^2{\displaystyle \frac{\theta }{2}}+2\sin {\displaystyle \frac{\theta }{2}}\cos {\displaystyle \frac{\theta }{2}}}{2{\cos }^2{\displaystyle \frac{\theta }{2}}+2\sin {\displaystyle \frac{\theta }{2}}\cos {\displaystyle \frac{\theta }{2}}}}(\mathrm{升}\mathrm{幂}\mathrm{降}\mathrm{角})=\tan {\displaystyle \frac{\theta }{2}}$
$2.\mathrm{利}\mathrm{用}\mathrm{利}\mathrm{用}1\pm \sin \theta$
$\mathrm{原}\mathrm{式}={\displaystyle \frac{(\sin {\displaystyle \frac{\theta }{2}}+\cos {\displaystyle \frac{\theta }{2}}{)}^2-({\cos }^2{\displaystyle \frac{\theta }{2}}-{\sin }^2{\displaystyle \frac{\theta }{2}})}{(\sin {\displaystyle \frac{\theta }{2}}+\cos {\displaystyle \frac{\theta }{2}}{)}^2+({\cos }^2{\displaystyle \frac{\theta }{2}}-{\sin }^2{\displaystyle \frac{\theta }{2}})}}(\mathrm{提}\mathrm{公}\mathrm{因}\mathrm{式})$

方法一：
$\mathrm{直}\mathrm{接}\mathrm{算}\mathrm{出}\sin {15}^{\circ },\cos {15}^{\circ }\mathrm{代}\mathrm{入}$
方法二：
$(\mathrm{升}\mathrm{幂})\mathrm{原}\mathrm{式}={\displaystyle \frac{(\sin {15}^{\circ }+\cos {15}^{\circ }{)}^2}{{\sin }^2{15}^{\circ }-{\cos }^2{15}^{\circ }}}={\displaystyle \frac{1+\sin {30}^{\circ }}{\cos {30}^{\circ }}}=\sqrt{3}$
方法三：
$(\mathrm{化}\mathrm{为}\mathrm{正}\mathrm{切})\mathrm{原}\mathrm{式}={\displaystyle \frac{\tan {15}^{\circ }+1}{\tan {15}^{\circ }-1}}=-{\displaystyle \frac{\tan {15}^{\circ }+\tan {45}^{\circ }}{1-\tan {15}^{\circ }\tan {45}^{\circ }}}=\tan ({15}^{\circ }+{45}^{\circ })=\sqrt{3}$
方法四：
$(\mathrm{变}\mathrm{角})\mathrm{原}\mathrm{式}={\displaystyle \frac{(\sin {45}^{\circ }-{30}^{\circ })+\sin ({45}^{\circ }+{30}^{\circ })}{(\sin {45}^{\circ }-{30}^{\circ })-\sin ({45}^{\circ }+{30}^{\circ })}}(\mathrm{展}\mathrm{开}\mathrm{即}\mathrm{可})$
变角的方式有多种

$\tan {\displaystyle \frac{\theta }{2}}={\displaystyle \frac{\sin {\displaystyle \frac{\theta }{2}}}{\cos {\displaystyle \frac{\theta }{2}}}}={\displaystyle \frac{{\sin }^2{\displaystyle \frac{\theta }{2}}}{\cos {\displaystyle \frac{\theta }{2}}\sin {\displaystyle \frac{\theta }{2}}}}={\displaystyle \frac{2{\sin }^2{\displaystyle \frac{\theta }{2}}}{2\cos {\displaystyle \frac{\theta }{2}}\sin {\displaystyle \frac{\theta }{2}}}}={\displaystyle \frac{1-\cos \theta }{\sin \theta }}$

$\mathrm{设}\alpha ={\displaystyle \frac{\alpha +\beta }{2}}+{\displaystyle \frac{\alpha -\beta }{2}},\beta ={\displaystyle \frac{\alpha +\beta }{2}}-{\displaystyle \frac{\alpha -\beta }{2}}$
$\sin \alpha +\sin \beta =\sin ({\displaystyle \frac{\alpha +\beta }{2}}+{\displaystyle \frac{\alpha -\beta }{2}})+\sin ({\displaystyle \frac{\alpha +\beta }{2}}-{\displaystyle \frac{\alpha -\beta }{2}})\mathrm{展}\mathrm{开}\mathrm{即}\mathrm{可}$