If $f(z) + \sin(z)$ is an analytic function on a domain D and $f(z) + \cos(z)$ is analytic on D, then $f(z)$ is constant on D. Is this true?
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Why would this be true? Since $\cos$ and $\sin$ are both analytic functions, if you take any analytic function $f$, the functions $f+\sin$ and $f+\cos$ will both be analytic.