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This answer is partially correct. You didn't explain why the energy increases with excited states. It is because, the wavefunction corresponding to the higher excited states have more nodes. More nodes increases the discontinuity of the wavefunction that is del square term in the Schrodingers equation.
Can't just get the ans. .. is there a wavefunction behaving both sine and cosine for a symmetric potential box ! .. I think this can't be possible .. cause existence of both sine and cosine needs an intermediate .. but only discrete values are allowed .. so can u make the ans .. to one line .. thank you .. your work is great .