Question

Нужно доказать тригонометрическое тождество: 1+ctg a\1+tg a=ctg a

Answer 1

$\frac{1+ctg \alpha }{1+tg \alpha } = \frac{1+ \frac{cos \alpha }{sin \alpha }}{1+ \frac{sin \alpha }{cos \alpha } } = \\ = \frac{\frac{cos \alpha +sin \alpha }{sin \alpha } }{ \frac{sin \alpha +cos \alpha }{cos \alpha } } = \frac{cos \alpha +sin \alpha }{sin \alpha } * \frac{cos \alpha }{sin \alpha +cos \alpha} = \\ = \frac{cos \alpha}{sin \alpha}=ctg \alpha$

Answer 2

$\frac{1+ctga}{1+tga} = \frac{1+ \frac{cosx}{sina} }{1+ \frac{sina}{cosa} } = \frac{ \frac{cos ^{2}x+sin ^{3} +cosa }{sina} }{ \frac{cos ^{3}x+sin ^{2} a+sina }{cosa} } = \frac{(cos ^{2}x+sin ^{3} +cosa)*cosa}{(cos ^{3}x+sin ^{2} a+sina)*sina} =ctga$