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Solution for $n=2$: In view of the rotational invariance, the distribution of the random point $(C,S)$ is the same as that of $(1,0)+(\cos U,\sin U)=(1+\cos U,\sin U)$, where $U$ is uniformly distributed on $[0,2\pi]$. So, $R$ is equal in distribution to $\sqrt{(1+\cos U)^2+\sin^2 U}=2|\cos U/2|$ and hence to $2\cos V$, where $V$ is uniformly distributed on $[0,\pi/2]$. So, the pdf $f_R$ of $R$ is given by $$f_R(r)=\frac2{\pi\sqrt{4-r^2}}1_{0<r<2} $$ for real $r$.