How do you simplify (cos^2x-4)/(cos^2x-2)?

Nov 6, 2015

$\frac{4}{3 - \cos \left(2 x\right)} + 1$

Explanation:

$\frac{{\cos}^{2} \left(x\right) - 4}{{\cos}^{2} \left(x\right) - 2} = \frac{2 + \left[2 - {\cos}^{2} \left(x\right)\right]}{2 - {\cos}^{2} \left(x\right)}$

$= \frac{2}{2 - {\cos}^{2} \left(x\right)} + 1$

$= \frac{4}{4 - 2 {\cos}^{2} \left(x\right)} + 1$

$= \frac{4}{3 - \cos \left(2 x\right)} + 1$,

from the double angle identity $\cos \left(2 \theta\right) \equiv 2 {\cos}^{2} \left(\theta\right) - 1$.