Udowodnij, że ctg(10)*ctg(50)*ctg(70) = 1,732...

[igor123]

Udowodnij, że \(\displaystyle{ ctg(10) ctg(50) ctg(70) = \sqrt{3}}\)

[bartholdy]

\(\displaystyle{ L = ctg 10^\circ ctg (60^\circ - 10^\circ) ctg (60^\circ + 10^\circ) = ctg 10^\circ \frac{ctg 60^\circ ctg 10^\circ + 1}{ctg 10^\circ - ctg 60^\circ}\cdot\frac{ctg 60^\circ ctg 10^\circ - 1}{ctg 10^\circ + ctg 60^\circ} = \frac{ctg 10^\circ ( ctg^2 60^\circ\cdot ctg^2 10^\circ - 1 )}{ctg^2 10^\circ - ctg^2 60^\circ} = \frac{ctg 10^\circ ( ctg^210^\circ - 3 )}{3 ctg^2 10^\circ - 1} = ctg30^\circ = \sqrt{3} = P}\)