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At a crude level, the evaluator that does the TrigExpand can be emulated by a FixedPoint or ReplaceRepeated. If we want to see the steps behind this particular kind of evaluation, it seems possible that a simple substitute for the built-in TrigExpand can be implemented based on FixedPointList instead of FixedPoint, to keep track of the steps. Here is a ...


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Because you made a mistake when using double angle identity. In your equations, I got(2 Sin[θ/2]) == (1 - Cos[θ]), where it should be (2 Sin[θ/2]^2) == (1 - Cos[θ]) Assuming[μ > 0 && g > 0 && ℏ > 0 && m > 0 && k > 0, Integrate[(4 μ^2 g^2)/(ℏ^4 (m^2 + 4 k^2 Sin[θ/2]^2)^2) Sin[θ], {θ, 0, π}, {ϕ, 0, 2 π}]] // ...



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