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Ответ:

а)

[tex]1 - \cos {}^{2} ( \alpha ) + {tg}^{2} \alpha \times \cos {}^{2} ( \alpha ) = \\ = \sin {}^{2} ( \alpha ) + \frac{ \sin {}^{2} ( \alpha ) }{ \cos {}^{2} ( \alpha ) } \times \cos {}^{2} ( \alpha ) = \\ = \sin {}^{2} ( \alpha ) + \sin {}^{2} ( \alpha ) = 2 \sin {}^{2} ( \alpha ) [/tex]

б)

[tex] \sqrt{1 - \sin {}^{2} ( \beta ) } = \sqrt{ \cos {}^{2} ( \beta ) } = \cos( \beta ) \\ [/tex]

в)

[tex] \frac{1}{ 1 + \cos( \alpha ) } + \frac{1}{1 - \cos( \alpha ) } = \frac{1 - \cos( \alpha ) + 1 + \cos( \alpha ) }{(1 + \cos( \alpha ) )(1 - \cos( \alpha )) } = \\ = \frac{2}{1 - \cos {}^{2} ( \alpha ) } = \frac{2}{ \sin {}^{2} ( \alpha ) } [/tex]

г)

[tex]tg \alpha \times ctg \alpha + \frac{tg \alpha }{ctg \alpha } = 1 + tg \alpha \times tg \alpha = \\ = 1 + {tg}^{2} \alpha = \frac{1}{ \cos {}^{2} ( \alpha ) } [/tex]