Question

Помогите пожалуйста дам 80 балл. Алгебра:) ​

Answer 1

22.5

1.

$\frac{ \sin(x) + \cos(x) }{1 + tgx} = \frac{ \sin(x) + \cos(x) }{1 + \frac{ \sin(x) }{ \cos(x) } } = \\ = ( \sin(x) + \cos(x)) \times \frac{ \cos(x) }{ \sin(x) + \cos(x) } = \\ = \cos(x)$

2.

$\frac{ctgx - 1}{ \sin(x) - \cos(x) } = \frac{ \frac{ \cos(x) }{ \sin(x) } - 1 }{ \sin(x) - \cos(x) } = \\ = \frac{ \cos(x) - \sin(x) }{ \sin(x) } \times \frac{1}{ \sin(x) - \cos(x) } = \\ = \frac{ - ( \sin(x) - \cos(x)) }{ \sin(x) \times ( \sin(x) - \cos(x)) } = \\ = - \frac{1}{ \sin(x) }$

3.

$\frac{1 + ctgx}{ \sin(x) + \cos(x) } = \frac{1 + \frac{ \cos(x) }{ \sin(x) } }{ \sin(x) + \cos(x) } = \\ = \frac{ \sin(x) + \cos(x) }{ \sin(x) } \times \frac{1}{ \sin(x) + \cos(x) } = \\ = \frac{1}{ \sin(x) }$

4.

$\frac{ \sin(x) - \cos(x) }{1 - tgx} = \frac{ \sin(x) - \cos(x) }{1 - \frac{ \sin(x) }{ \cos(x) } } = \\ = ( \sin(x) - \ cos(x) ) \times \frac{ \cos(x) }{ \cos(x) - \sin(x) } = \\ = \frac{ - ( \sin(x) - \cos(x) ) \times \cos(x) }{ \sin(x) - \cos(x) } = - \cos(x)$

22.6

1.

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

2.

${ \sin }^{2} \gamma + \frac{ {ctg}^{2} \gamma - 1}{ {ctg}^{2} \gamma + 1 } = { \sin }^{2} \gamma + \frac{ \frac{ { \cos}^{2} \gamma }{ { \sin}^{2} \gamma } - 1 }{ \frac{ { \cos}^{2} \gamma }{ { \sin }^{2} \gamma } + 1 } = \\ = { \sin }^{2} \gamma + \frac{ { \cos }^{2} \gamma - { \sin}^{2} \gamma }{ { \sin }^{2} \gamma } \times \frac{ { \sin}^{2} \gamma }{ { \sin }^{2} \gamma + { \cos }^{2} \gamma } = \\ = { \sin}^{2} \gamma + { \cos }^{2} \gamma - { \sin }^{2} \gamma = { \cos }^{2} \gamma$

3.

$\frac{ {ctg}^{2} \gamma - 1}{ {ctg}^{2} \gamma + 1 } - { \cos }^{2} \gamma = \frac{ \frac{ { \cos }^{2} \gamma }{ { \sin}^{2} \gamma } - 1}{ \frac{ { \cos }^{2} \gamma }{ { \sin }^{2} \gamma } + 1 } - { \cos}^{2} \gamma = \\ = \frac{ { \cos }^{2} \gamma - { \sin }^{2} \gamma }{ { \sin }^{2} \gamma } \times \frac{ { \sin }^{2} \gamma }{ { \cos}^{2} \gamma + { \sin }^{2} \gamma } - { \cos }^{2} \gamma = \\ = { \cos }^{2} \gamma - { \sin }^{2} \gamma - { \cos }^{2} \gamma = \\ = - { \sin }^{2} \gamma$

4.

$\frac{ {tg}^{2}x - 1 }{ {tg}^{2}x + 1 } - { \sin}^{2} x = \frac{ \frac{ { \sin}^{2}x }{ { \cos }^{2}x } - 1}{ \frac{ { \sin }^{2}x }{ { \cos }^{2} x} + 1} - { \sin}^{2} x = \\ = \frac{ { \sin }^{2}x - { \cos }^{2}x }{ { \cos }^{2}x } \times \frac{ { \cos }^{2} x}{ { \sin }^{2}x + { \cos }^{2} x } - { \sin }^{2} x = \\ = { \sin }^{2} x - { \cos }^{2} x - { \sin }^{2} x = - { \cos }^{2} x$