Question

arctg x+arctg y=pi/2. Sa se arate ca x*y=1

Answer 1

$ctg(arctgx+arctgy)=\dfrac{1}{tg(arctgx+arctgy)}=\dfrac{1}{\dfrac{x+y}{1-xy}}=\dfrac{1-xy}{x+y}.$

Pe de altă parte:

$ctg\left(\dfrac{\pi}{2}\right)=\dfrac{cos\left(\dfrac{\pi}{2}\right)}{sin\left(\dfrac{\pi}{2}\right)}=\dfrac{0}{1}=0.$

Deci 1-xy = 0, de aici xy = 1, ceea ce trebuia demonstrat.

Green eyes.