Question

упростите выражение ​

Answer 1

Ответ:

1.

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

2.

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

3.

$\frac{tg \alpha - tg \beta }{ctg \alpha + ctg \beta } = \frac{ \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \frac{ \sin( \beta ) }{ \cos( \beta ) } }{ \frac{ \cos( \alpha ) }{ \sin( \alpha ) } + \frac{ \cos( \beta ) }{ \sin( \beta ) } } = \\ = \frac{ \sin( \alpha ) \cos( \beta ) - \sin( \beta ) \cos( \alpha ) }{ \cos( \alpha ) \cos( \beta ) } \times \frac{ \sin( \alpha ) \sin( \beta ) }{ \cos( \alpha ) \sin( \beta ) + \cos( \beta ) \sin( \alpha ) } = \\ = \frac{ \sin( \alpha - \beta ) \times \sin( \alpha ) \sin( \beta ) }{ \cos( \alpha ) \cos( \beta ) \times \sin( \beta + \alpha ) } = \frac{ \sin( \alpha - \beta ) }{ \sin( \alpha + \beta ) } \times tg \alpha \times tg \beta$