Can we prove this trig identity to solve this equation?

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Prove that \[ \frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = \frac{2}{\sin A} \] for all values of \(A\).

Find the values of \(x\) between \(0^\circ\) and \(360^\circ\) if \[ \frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x} = 1 + 3 \sin x. \]