Question

Упростите:
tga/sina-sina/ctga-cosa
$\frac{ \tan( \alpha ) }{ \sin( \alpha ) } - \frac{ \sin( \alpha ) }{ \cot( \alpha ) } - \cos( \alpha )$
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Answer 1

$\frac{tg \, \alpha}{\sin{\alpha}}-\frac{\sin{\alpha}}{ctg \, {\alpha}}-\cos{\alpha}=\frac{\frac{\sin{\alpha}}{\cos{\alpha}}}{\sin{\alpha}}-\frac{\sin{\alpha}}{\frac{\cos{\alpha}}{\sin{\alpha}}}-\cos{\alpha}=\frac{\sin {\alpha}}{\sin {\alpha}\cdot \cos{\alpha}}-\frac{\sin{\alpha}\cdot \sin{\alpha}}{\cos{\alpha}}-\cos{\alpha}= \\ \\ =\frac{1}{\cos{\alpha}}-\frac{\sin^2{\alpha}}{\cos{\alpha}}-\cos{\alpha}=\frac{1}{\cos{\alpha}}-\frac{1-\cos^2{\alpha}}{\cos{\alpha}}-\cos{\alpha}=$

$=\frac{1}{\cos{\alpha}}-(\frac{1}{\cos{\alpha}}-\frac{\cos^2{\alpha}}{\cos{\alpha}})-\cos{\alpha}=\frac{\cos^2{\alpha}}{\cos{\alpha}}-\cos{\alpha}=\cos{\alpha}-\cos{\alpha}=0$