This is the soviet problem that inspired me. Perhaps by solving this one, there will be insight to the solution of the other one.

Find the sum of the values, in degrees, that satisfy \(x\) from \( [0^{\circ}, 360^{\circ}]\) in the equation \(\displaystyle \sin^{5} x - \cos^{5} x = \frac{1}{\cos x} - \frac{1}{\sin x} \)

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