Question

Мне нужна тот кто регить +5 звезд

Answer 1

1.

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

2.

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

3.

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

4.

$4 - tg \alpha \times ctg \alpha = 4 - 1 = 3$

5.

$\cos( \beta ) - \sin {}^{2} ( \beta ) \cos {}^{2} ( \beta ) = \cos {}^{2} ( \beta ) \times (1 - \sin {}^{2} ( \beta )) = \\ = \cos {}^{2} ( \beta ) \times \cos {}^{2} ( \beta ) = \cos {}^{4} ( \beta )$

6.

$\sin {}^{4} ( \beta ) + \sin { }^{2} ( \beta ) \cos {}^{2} ( \beta ) = \sin {}^{2} ( \beta ) ( \sin {}^{2} ( \beta ) + \cos {}^{2} ( \beta ) ) = \\ = \sin {}^{2} ( \beta ) \times 1 = \sin {}^{2} ( \beta )$

7.

${tg}^{2} ( \beta )ctg {}^{2} ( \beta ) - \sin {}^{2} ( \beta ) = 1 - \sin {}^{2} ( \beta ) = \cos {}^{2} ( \beta ) \\$

8.

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

9.

$\frac{ \cos( \alpha ) + ctg \alpha }{1 + \sin( \alpha ) } = \frac{ \cos( \alpha ) + \frac{ \cos( \alpha ) }{ \sin( \alpha ) } }{ 1 + \sin( \alpha ) } = \\ = \frac{ \cos( \alpha ) \sin( \alpha ) + \cos( \alpha ) }{ \sin( \alpha ) } \times \frac{1}{1 + \sin( \alpha ) } = \\ = \frac{ \cos( \alpha ) ( \sin( \alpha ) + 1) }{ \sin( \alpha ) ( \sin( \alpha ) + 1) } = ctg \alpha$

10.

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

11.

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

12.

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

13.

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

14.

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

15.

$\frac{1 + {ctg}^{4} \alpha }{ {tg}^{2} \alpha + ctg {}^{2} \alpha } = \\ = (1 + {ctg}^{4} \alpha ) \times \frac{1}{ {ctg}^{2} \alpha + \frac{1}{ {ctg}^{2} \alpha } } = \\ = (1 + {ctg}^{4} \alpha ) \times \frac{ {ctg}^{2} \alpha }{1 + {ctg}^{4} \alpha } = {ctg}^{2} \alpha$

16.

$\frac{ \cos {}^{2} ( \alpha ) }{1 + {tg}^{2} \alpha } - \frac{ \sin {}^{2} ( \alpha ) }{ 1 + {ctg}^{2} \alpha } = \\ = \frac{ \cos {}^{2} ( \alpha ) }{1 + \frac{ \sin {}^{2} ( \alpha ) }{ \cos {}^{2} ( \alpha ) } } - \frac{ \sin {}^{2} ( \alpha ) }{1 + \frac{ \cos {}^{2} ( \alpha ) }{ \sin {}^{2} ( \alpha ) } } = \\ = \cos {}^{2} ( \alpha ) \times \frac{ \cos {}^{2} ( \alpha ) }{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) } - \sin {}^{2} ( \alpha ) \times \frac{ \sin {}^{2} ( \alpha ) }{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) } = \\ = \cos {}^{4} ( \alpha ) - \sin {}^{4} ( \alpha ) = \\ = ( \cos {}^{2} ( \alpha ) - \sin {}^{2} ( \alpha ) ) (\cos {}^{2} ( \alpha ) + \sin {}^{2} ( \alpha )) = \cos(2 \alpha )$