Question

лямбда[email protected]
[email protected]/1+ctg [email protected]/[email protected] = [email protected] + ctg @ - 1

Answer 1

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

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

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