Is $x(t) = \cos t + \sin\left(\frac{1}{2}t\right)$ 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.
As for every $t_0\in\mathbb{R}$ and $k\in\mathbb{Z}$
$$
\begin{eqnarray}
  &amp;x(t_0+4k\pi)  &amp;=\cos(t_0+4k\pi)+\sin(t_0/2+2k\pi)\\
  &amp;              &amp;=\cos(t_0)+\sin(t_0/2)\\
  &amp;              &amp;=x(t_0)
\end{eqnarray}
$$
you answer is correct: $x(t)$ is periodic.