Question

помогите пожалуйста
​

Answer 1

Ответ:

1)

$\frac{2}{ { \sin( \beta ) }^{2} }$

2) 1

3)

$\frac{2}{ \cos( \beta ) }$

4)

$\frac{1}{ \cos(x) }$

Объяснение:

1)

${(1 + \cot( \beta )) }^{2} + {( 1- \cot( \beta ) )}^{2} = \\ = 1 + 2 \cot( \beta ) + { \cot( \beta ) }^{2} + 1 - 2 \cot( \beta ) + { \cot( \beta ) }^{2} = \\ = 2 + 2 { \cot( \beta ) }^{2} = \\ = 2(1 + { \cot( \beta ) }^{2} ) = 2 \times \frac{1}{ { \sin( \beta ) }^{2} } = \\ = \frac{2}{ { \sin( \beta ) }^{2} }$

2)

${ \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} ( { \tan( \alpha ) }^{2} + { \cot( \alpha ) }^{2} + 2) = \\ = { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} ( \frac{ { \sin( \alpha ) }^{2} }{ { \cos( \alpha ) }^{2} } + \frac{ { \cos( \alpha ) }^{2} }{ { \sin( \alpha ) }^{2} } + 2) = \\ = { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} ( \frac{ { \sin( \alpha ) }^{4} + { \cos( \alpha ) }^{4} }{ { \cos( \alpha ) }^{2} { \sin( \alpha ) }^{2} } + 2) = \\ = { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} \times \frac{ { \sin( \alpha ) }^{4} + { \cos( \alpha ) }^{4} }{ { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} } + 2 { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} = \\ = { \sin( \alpha ) }^{4} + 2 { \sin( \alpha ) }^{2} { \cos( \alpha ) }^{2} + { \cos( \alpha ) }^{4} = {( { \sin( \alpha ) }^{2} + { \cos( \alpha ) }^{2}) }^{2} = \\ = {1}^{2} = 1$

3)

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

4)

$\tan(x) + \frac{ \cos(x) }{1 + \sin(x) } = \\ = \frac{ \tan(x)(1 + \sin(x) ) + \cos(x) }{1 + \sin(x) } = \\ = \frac{ \frac{ \sin(x) }{ \cos(x) } (1 + \sin(x) ) + \cos(x) }{1 + \sin(x) } = \\ = \frac{ \frac{ \sin(x) }{ \cos(x) } + \frac{ \sin(x) }{ \cos(x) } \times \sin(x) + \cos(x) }{1 + \sin(x) } = \\ = \frac{ \frac{ \sin(x) }{ \cos(x) } + \frac{ { \sin(x) }^{2} }{ \cos(x) } + \cos(x) }{1 + \sin(x) } = \\ = \frac{ \frac{ \sin(x) + { \sin(x) }^{2} + { \cos(x) }^{2} }{ \cos(x) } }{1 + \sin(x) } = \\ = \frac{ \sin(x) + 1}{ \cos(x) (1 + \sin(x) )} = \frac{1}{ \cos(x) }$