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trig function differentiable at 0, f(x)=g(x)f(x/2) recurrence

Source: Bulgaria 1973 P4

June 20, 2021
trigonometryfunctionfefunctional equationalgebra

Problem Statement

Find all functions f(x)f(x) defined in the range (−π2,π2)\left(-\frac\pi2,\frac\pi2\right) that are differentiable at 00 and satisfy f(x)=12(1+1cos⁡x)f(x2)f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right) for every xx in the range (−π2,π2)\left(-\frac\pi2,\frac\pi2\right).
L. Davidov
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