Question

Упростите выражение ​

Answer 1

Ответ:

1.

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

2.

$ctg \beta + \frac{ \sin( \beta ) }{1 + \cos( \beta ) } = \\ = \frac{ \cos( \beta ) }{ \sin( \beta ) } + \frac{ \sin( \beta ) }{ 1 + \cos( \beta ) } = \\ = \frac{ \cos( \beta ) (\cos( \beta ) + 1) + \sin ^{2} ( \beta ) }{ \sin( \beta ) (1 + \cos( \beta )) } = \\ = \frac{ { \cos}^{2} \beta + \cos( \beta ) + { \sin }^{2} \beta }{ \sin( \beta ) (1 + \cos( \beta )) } = \frac{1 + \cos( \beta ) }{ \sin( \beta ) ( \cos( \beta ) + 1) } = \\ = \frac{1}{ \sin( \beta ) }$