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Here is a real solution for $k=2$: Take $f_1(x)=x$ and $$f_2(x)=\arccos\left( \sqrt{\tfrac{3}{4}- \cos ^4(x/2)} \operatorname{cotan} (x/2)-\tfrac{1}{2}\cos x\right).$$ Then $$\cos f_1(x)+\cos f_2(x)=\cos\bigl(f_1(x)+f_2(x)\bigr),\;\;\text{for}\;\;x_{\rm min}<x<x_{\rm max}.$$ Numerically, I find $x_{\rm min}=1.05$ and $x_{\rm max}=5.53$.