Question

плиз сделайте пожалуйста побыстрее ​

Answer 1

Ответ:

3

$ctg \gamma - \sin( 2\gamma ) - ctg \gamma \cos( 2\gamma ) = \\ = ctg \gamma (1 - \cos( 2\gamma )) - \sin( 2\gamma ) = \\ = ctg \gamma (1 - \cos {}^{2} ( \gamma ) + \sin {}^{2} ( \gamma ) ) - \sin(2 \gamma )) = \\ = \frac{ \cos( \gamma ) }{ \sin( \gamma ) } \times 2 \sin {}^{2} ( \gamma ) - \sin( 2\gamma ) = \\ = 2 \sin( \gamma ) \cos( \gamma ) - 2 \sin( \gamma ) \cos( \gamma ) = 0$

4

$\frac{1}{1 - tg \alpha } - \frac{1}{1 + tg \alpha } = \frac{1}{1 - \frac{ \sin( \alpha ) }{ \cos( \alpha ) } } - \frac{1}{1 + \frac{ \sin( \alpha ) }{ \cos( \alpha ) } } = \\ = \frac{ \cos( \alpha ) }{ \cos( \alpha ) - \sin( \alpha ) } - \frac{ \cos( \alpha ) }{ \sin( \alpha ) + \sin( \alpha ) } = \\ = \frac{ \cos( \alpha )( \sin( \alpha ) + \cos( \alpha )) - \cos( \alpha ) (\cos( \alpha ) - \sin( \alpha )) }{ (\cos( \alpha ) - \sin( \alpha ) ) (\cos( \alpha ) + \sin( \alpha )) } = \\ = \frac{ \cos( \alpha ) ( \sin( \alpha ) + \cos( \alpha ) - \cos( \alpha ) + \sin( \alpha ) )}{ \cos {}^{2} ( \alpha ) - \sin {}^{2} ( \alpha ) } = \\ = \frac{ \cos( \alpha ) \times 2 \sin( \alpha ) }{ \cos(2 \alpha ) } = \frac{ \sin( 2\alpha ) }{ \cos( 2\alpha ) } = tg2 \alpha$

5

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

6

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

7

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

8

$1 + \frac{ \cos( \alpha ) }{ \sin( \alpha ) } - \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - 2ctg2 \alpha = \\ = 1 + \frac{ \cos {}^{2} ( \alpha ) - \sin {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - 2ctg 2\alpha = \\ = 1 + \frac{ \cos(2 \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - 2 \times \frac{ \cos( 2\alpha ) }{ \sin(2 \alpha ) } = \\ = 1 + \frac{2 \cos( 2\alpha ) }{2 \sin( \alpha ) \cos( \alpha ) } - \frac{2 \cos( 2\alpha ) }{ \sin(2 \alpha ) } = \\ = 1 + \frac{2 \cos( 2\alpha ) }{ \sin( 2\alpha ) } - \frac{2 \cos(2 \alpha ) }{ \sin( 2\alpha ) } = 1$

9

$\cos {}^{2} ( \alpha ) - 4 \sin {}^{2} ( \frac{ \alpha }{2} ) \cos {}^{2} ( \frac{ \alpha }{2} ) - \cos( 2\alpha ) = \\ = \cos {}^{2} ( \alpha ) - {(2 \sin( \frac{ \alpha }{2} ) \cos( \frac{ \alpha }{2} ) )}^{2} - \cos {}^{2} ( \alpha ) + \sin {}^{2} ( \alpha ) = \\ = \sin {}^{2} ( \alpha ) - \sin {}^{2} ( \alpha ) = 0$

10

$ctg2 \alpha - ctg \alpha = \frac{ \cos( 2\alpha ) }{ \sin( 2\alpha ) } - \frac{ \cos( \alpha ) }{ \sin( \alpha ) } = \\ = \frac{ \cos( 2\alpha ) \sin( \alpha ) - \cos( \alpha ) \sin( 2\alpha ) }{ \sin(2 \alpha ) \sin( \alpha ) } = \\ = \frac{ \sin( \alpha - 2\alpha ) }{ \sin( 2\alpha ) \sin( \alpha ) } = \frac{ - \sin( \alpha ) }{ \sin( 2\alpha ) \sin( \alpha ) } = \\ = - \frac{1}{ \sin(2 \alpha ) }$