Question

Найдите значение выражения tg³a - ctg3a, если tga - ctga = t.

Answer 1

$\displaystyle\bf\\tg\alpha -Ctg\alpha =t\\\\(tg\alpha -Ctg\alpha)^{2} =t^{2} \\\\tg^{2} \alpha -2\cdot \underbrace{tg\alpha \cdot Ctg\alpha}_{1} +Ctg^{2} \alpha=t^{2} \\\\tg^{2} \alpha -2+Ctg^{2} \alpha =t^{2} \\\\tg^{2} \alpha +Ctg^{2} \alpha =t^{2} +2\\\\\\tg^{3} \alpha -Ctg^{3} \alpha =(tg\alpha -Ctg\alpha )\cdot(tg^{2} \alpha +tg\alpha \cdot Ctg\alpha +Ctg^{2} \alpha )=\\\\\\=t\cdot(t^{2} +2+1)=t\cdot(t^{2}+3)$