Why does the Bohr-Sommerfeld rule for quantization give the exact energy-levels for a simple harmonic oscillator?
One can use various kinds of supersymmetry to argue that the WKB approximation for the quantum harmonic oscillator is exact. One method uses localization of path integrals, cf. e.g. Ref. 1. Another method uses supersymmetric quantum mechanics, cf. e.g. Ref. 2.
It is one of rare cases when an approximate formula coincides with the exact one. Another example is the Plank formula obtained originally as a bridging of two experimental asymptotes. A fluke. There are some other cases of happy coincidents.