Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$.
Let $\mu_1,\mu_2$ be its characteristic values. Then, a known theorem gives: $$\mu_1\mu_2=\exp\Bigg\{\int_0^{2\pi} tr(A(t))dt\Bigg\} \iff \mu_1\mu_2=e^{4\pi} $$
Therefore, $\mu_1,\mu_2$ are solutions of a quadratic equation of real coefficients that has the form: $$ \mu^2-\varphi\mu+e^{4\pi}=0 \quad (1) $$
For at least one solution to be unbounded, the discriminant of $(1)$ must be non-negative. $$ \Delta=\varphi^2-4e^{4\pi} \geq 0 \iff |\varphi|\geq 2e^{2\pi}$$
How can I show that this condition for $\varphi$ is satisfied?