[tex]\displaystyle \sf tg^3(x)-2tg^2(x)-tg(x)+2= 0 \\\\\ tg^2(x)\cdot \left[tg(x)-2\right] -\left[tg(x)-2\right] = 0 \\\\ \left[tg(x)-2\right]\cdot [tg^2(x)-1] = 0 \\\\ tg(x) - 2=0\to tg(x)=2 \\\\ \boxed{\sf x=Arc\ tg(2) \ \ ; \ \ x = \pi +Arc\ tg(2) }(\text{Duas solu\c c\~oes}) \\[/tex]
[tex]\displaystyle \sf tg^2(x) -1 = 0 \to tg(x) = \pm 1 \\\\ tg(x) = 1 \to \boxed{\sf \ x = \frac{\pi}{4\ } \ \ ;\ \ x = \frac{5\pi }{4}}(\text{Duas solu\c c\~oes} ) \\\\\\ tg(x) =-1 \to \boxed{\sf\ x=\frac{3\pi}{4} \ \ ; \ \ x = \frac{7\pi }{4} \ }(\text{Duas solu\c c\~oes} )[/tex]
Em todo os casos aparecem duas soluções distintas, totalizando 6 soluções distintas
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[tex]\displaystyle \sf tg^3(x)-2tg^2(x)-tg(x)+2= 0 \\\\\ tg^2(x)\cdot \left[tg(x)-2\right] -\left[tg(x)-2\right] = 0 \\\\ \left[tg(x)-2\right]\cdot [tg^2(x)-1] = 0 \\\\ tg(x) - 2=0\to tg(x)=2 \\\\ \boxed{\sf x=Arc\ tg(2) \ \ ; \ \ x = \pi +Arc\ tg(2) }(\text{Duas solu\c c\~oes}) \\[/tex]
[tex]\displaystyle \sf tg^2(x) -1 = 0 \to tg(x) = \pm 1 \\\\ tg(x) = 1 \to \boxed{\sf \ x = \frac{\pi}{4\ } \ \ ;\ \ x = \frac{5\pi }{4}}(\text{Duas solu\c c\~oes} ) \\\\\\ tg(x) =-1 \to \boxed{\sf\ x=\frac{3\pi}{4} \ \ ; \ \ x = \frac{7\pi }{4} \ }(\text{Duas solu\c c\~oes} )[/tex]
Em todo os casos aparecem duas soluções distintas, totalizando 6 soluções distintas
letra (d)