# Find x where pi <= x <= 2pi? 4 cos^2 x - 2= sec^2 x - tan^2 x

May 23, 2018

(4pi)/3; (11pi)/6

#### Explanation:

$4 {\cos}^{2} x - 2 = {\sec}^{2} x - {\tan}^{2} x$ (1)
Develop the right side:
$R S = \frac{1}{{\cos}^{2} x} - {\sin}^{2} \frac{x}{{\cos}^{2} x} = \frac{1 - {\sin}^{2} x}{{\cos}^{2} x} = {\cos}^{2} \frac{x}{{\cos}^{2} x} = 1$
The equation (1) becomes:
$4 {\cos}^{2} x - 2 = 1$
${\cos}^{2} x = \frac{3}{4}$
$\cos x = \pm \frac{\sqrt{3}}{2}$
a. $\cos x = \frac{\sqrt{3}}{2}$
Trig table and unit circle give 2 solutions of x -->
$x = \pm \frac{\pi}{6}$
b. $\cos x = - \frac{\sqrt{3}}{2}$ -->
$x = \pm \frac{2 \pi}{3}$
Inside the interval $\left(\pi , 2 \pi\right)$, the answers are:
$x = - \frac{\pi}{6}$, or $x = \frac{11 \pi}{6}$ (co-terminal)
$x = - \frac{2 \pi}{3}$, or $x = \pi + \frac{\pi}{3} = \frac{4 \pi}{3}$ (co-terminal)