I need some help with the following problem Let $t \mapsto\left(x_{1}(t), x_{2}(t)\right)$ be a solution of $$ \left\{\begin{array}{l} \dot{x_{1}}=x_{1}^{2}+\cos \left(t x_{1}\right)-1 \\ \dot{x_{2}}=\sin x_{2}+x_{1} \\ \left(x_{1}(0), x_{2}(0)\right)=(0,0) \end{array}\right. $$ Show that $x_{1}(t)=0=x_{2}(t)$ for all $t$.
Clearly, $x_{1}(t)=0=x_{2}(t)$ are trivial solution for given initial value problem for all $t$.
Now to show they are the only solution I am thinking to somehow show that for this Initial value problem there exists a unique solution, like using Picard's theorem. But I don't know how to use it for this case given the non-linear nature of the problem.