Лямбда[email protected] [email protected]/1+ctg [email protected]/[email protected] =...

Решите задачу:

$\frac{tg \alpha }{1+ctg \alpha } + \frac{ctg \alpha }{1+tg \alpha } = \frac{sin \alpha }{cos \alpha (1+ \frac{cos \alpha }{sin \alpha } )} + \frac{cos \alpha }{sin \alpha (1+ \frac{sin \alpha }{cos \alpha })} =\\\\= \frac{sin^2 \alpha }{cos \alpha (sin \alpha +cos \alpha )} + \frac{cos^2 \alpha }{sin \alpha (cos \alpha +sin \alpha )} =\\\\= \frac{sin^3 \alpha +cos^3 \alpha }{sin \alpha \cdot cos \alpha \cdot(sin \alpha +cos \alpha )} =$

$= \frac{(sin \alpha +cos \alpha )(sin^2 \alpha -sin \alpha \cdot cos \alpha +cos^2 \alpha )}{sin \alpha \cdot cos \alpha (sin \alpha +cos \alpha )} =$

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