Come risolvi ${sec}^{2} (x) - sec (x) = 2$ $sec ^ 2 (x) - sec (x) = 2$ ?

$x = π + 2 k π$ $x= pi + 2kpi$

$x = \frac{π}{3} + 2 k π$ $x = pi/3 + 2kpi$

$x = - \frac{π}{3} + 2 k π$ $x= -pi/3 + 2kpi$

Dal $sec (x) = \frac{1}{cos (x)}$ $sec(x)=1/cos(x)$ , l'espressione diventa

$\frac{1}{{cos}^{2} (x)} - \frac{1}{cos (x)} = 2$ $1/cos^2(x) - 1/cos(x) = 2$

Assumendo $cos (x) \neq 0$ $cos(x)ne 0$ , possiamo moltiplicare tutto per ${cos}^{2} (x)$ $cos^2(x)$ :

$1 - cos (x) = 2 {cos}^{2} (x)$ $1-cos(x) = 2cos^2(x)$ .

riorganizzare:

$2 {cos}^{2} (x) + cos (x) - 1 = 0$ $2cos^2(x)+cos(x)-1=0$ .

set $t = cos (x)$ $t=cos(x)$ :

$2 t^{2} + t - 1 = 0$ $2t^2+t-1=0$

Risolvi come al solito con la formula discriminante:

$t = - 1$ $t=-1$ , $t = \frac{1}{2}$ $t=1/2$

Converti le soluzioni:

$cos (x) = - 1 \Leftrightarrow x = π + 2 k π$ $cos(x)=-1 iff x=pi+2kpi$

$cos (x) = \frac{1}{2} \Leftrightarrow x = \pm \frac{π}{3} + 2 k π$ $cos(x)=1/2 iff x=pmpi/3 +2kpi$

Come risolvi sec ^ 2 (x) - sec (x) = 2 sec2(x)−sec(x)=2 sec ^ 2 (x) - sec (x) = 2 ?