Question

докажите тождество sin2a+sina/cos2a+1+cosa-1/ctga=0​

Answer 1

Ответ:

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