Is $x(t) = cos t + sinleft(frac{1}{2}tright)$ a periodic signal?

Is $x(t) = \cos t + \sin\left(\frac{1}{2}t\right)$ a periodic signal?

The answer provided by the book is different from my answer. Book says its not a periodic signal. Can you guys tell me why is it not a periodic signal?

My answer:

$\cos(t)$ is periodic as $2\pi f_1 = 1 \Rightarrow f_1=\frac{1}{2\pi} \Rightarrow T_1=2\pi$

$\sin\left(\frac{1}{2}t\right)$ is also periodic as $2\pi f_2 = \frac{1}{2} \Rightarrow f_2=\frac{1}{4\pi} \Rightarrow T_2=4\pi$

Therefore $\frac{T_1}{T_2} = \frac{2\pi}{4\pi} = \frac{1}{2}$ is rational number

Therefore the given $x(t)$ is a periodic signal.