tga+ctga

[profesorq]

Jesli widaomo ze \(\displaystyle{ tg\alpha + ctg\alpha=\sqrt{5}}\). to ile wynosi \(\displaystyle{ tg^8\alpha + ctg^8\alpha}\) ??

[baksio]

\(\displaystyle{ tg^8\alpha+ctg^8\alpha = (tg^4\alpha+ctg^4\alpha)^2 - 2tg^4\alpha*ctg^4\alpha = [(tg^2\alpha+ctg^2\alpha)^2 - 2]^2 -2}\)
Można teraz podnieść to równanie \(\displaystyle{ tg\alpha + ctg\alpha = \sqrt{5}}\) do kwadratu lub dalej rozpisać:
\(\displaystyle{ [(tg^2\alpha+ctg^2\alpha)^2 - 2]^2 -2 = ([(tg\alpha+ctg\alpha)^2-2]^2-2)^2 -2}\)