[tex] \frac{1 - \cos {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - \sin {}^{2} ( \alpha ) ( \tg( \alpha ) + \ctg( \alpha )) = \\ = \frac{ \sin {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - \sin {}^{2} ( \alpha ) ( \frac{ \sin( \alpha ) }{ \cos( \alpha ) } + \frac{ \cos( \alpha ) }{ \sin ( \alpha ) } ) = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \sin {}^{2} ( \alpha ) \times \frac{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \frac{ \sin {}^{2} ( \alpha ) }{ \sin ( \alpha ) \cos( \alpha ) } = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \frac{ \sin( \alpha ) }{ \cos( \alpha ) } = \tg( \alpha ) - \tg( \alpha ) = 0[/tex]
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[tex] \frac{1 - \cos {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - \sin {}^{2} ( \alpha ) ( \tg( \alpha ) + \ctg( \alpha )) = \\ = \frac{ \sin {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - \sin {}^{2} ( \alpha ) ( \frac{ \sin( \alpha ) }{ \cos( \alpha ) } + \frac{ \cos( \alpha ) }{ \sin ( \alpha ) } ) = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \sin {}^{2} ( \alpha ) \times \frac{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \frac{ \sin {}^{2} ( \alpha ) }{ \sin ( \alpha ) \cos( \alpha ) } = \\ = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } - \frac{ \sin( \alpha ) }{ \cos( \alpha ) } = \tg( \alpha ) - \tg( \alpha ) = 0[/tex]