Question

Ctg2•ctg4•...•ctg88=?

Answer 1

Stim urmatoarele relatii trigonometrice:
$\cos{(90-a)}-\sin{a}$
$\sin{(90-a)}=\cos{a}$
Atunci reiese ca:
$ctg(90-a)=\frac{\cos{(90-a)}}{\sin{(90-a)}}=\frac{-\sin{a}}{<span>\cos{a}</span>}=-tg(a)$ 
In general stim ca
$tg(a)*ctg(a)=1$
Atunci:
$ctg(a)*ctg(90-a)=ctg(a)*(-tg(a))=-(tg(a)*ctg(a))=-1$

Observam ca avem 44 de perechi de forma (a,90-a): (2,88),(4,86)...(44,46)
Atunci prin grupare o sa avem perechile de produse:
$ctg2*ctg(90-2)*ctg4*ctg(90-4)*...*ctg(44)*ctg(90-44)=(-1)^{44}=1$