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The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \sin x + a & = bx \\ \cos x &= b \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.

The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \cos x &= ax + b \\ \sin x + a &= 0 \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.