Example 4. Given $\sum_{j=1}^{n} a_{j} \cos \alpha_{j}=\sum_{j=1}^{n} a_{5} \cdot \cos \left(\alpha_{j}+1\right)=0$, find the value of $\sum_{j=1}^{n} a_{j} \cos \left(\alpha_{j}+\frac{\pi}{10}\right)$.