Question

daca tgx + ctg = 5 , atunci sa se calculeze :

a) $tg^{2} x$ + $ctg^{2} x$

b) $tg^{3} x + ctg^{3} x$

Answer 1

$\text{tg} x+\text{ctg}x = 5\Big|^2 \\ \\ \text{tg}^2 x+2\text{tg}x\text{ctg}x+\text{ctg}^2 x = 25 \\ \text{tg}^2 x+2+\text{ctg}^2 x = 25\\ \\\boxed{\text{tg}^2 x+\text{ctg}^2x = 23 } \Big|\cdot \text{tg} x+\text{ctg}x\\ \\ (\text{tg} x+\text{ctg}x )(\text{tg}^2 x+\text{ctg}^2x) = 23\cdot 5\\ \\ \text{tg}^3x+\text{tg} x\text{ctg}^2 x+\text{ctg} x\text{tg}^2x+\text{ctg}^3x = 115\\ \\ \text{tg}^3x+\text{ctg} x+\text{tg} x+\text{ctg}^3x = 115 \\ \\ \text{tg}^3x+5+\text{ctg}^3x = 115$

$\boxed{\text{tg}^3x+\text{ctg}^3x = 110}$