1. Fórmulas usadas:
$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$ y $\cot \alpha = \frac{\cos \alpha}{\sin \alpha}$.

2. Desarrollo paso a paso:
Analizamos el primer término:
$$ \frac{\tan^3 \alpha}{\sin^2 \alpha} = \frac{\sin^3 \alpha}{\cos^3 \alpha \cdot \sin^2 \alpha} = \frac{\sin \alpha}{\cos^3 \alpha} $$
Analizamos el tercer término:
$$ \frac{\cot^3 \alpha}{\cos^2 \alpha} = \frac{\cos^3 \alpha}{\sin^3 \alpha \cdot \cos^2 \alpha} = \frac{\cos \alpha}{\sin^3 \alpha} $$
Sumamos los términos modificados y el central:
$$ \frac{\sin \alpha}{\cos^3 \alpha} + \frac{\cos \alpha}{\sin^3 \alpha} - \frac{1}{\sin \alpha \cos \alpha} $$
Buscamos común denominador $\sin^3 \alpha \cos^3 \alpha$:
$$ \frac{\sin^4 \alpha + \cos^4 \alpha - \sin^2 \alpha \cos^2 \alpha}{\sin^3 \alpha \cos^3 \alpha} $$
Usando $\sin^4 \alpha + \cos^4 \alpha = (\sin^2 \alpha + \cos^2 \alpha)^2 - 2\sin^2 \alpha \cos^2 \alpha$:
$$ \frac{1 - 3\sin^2 \alpha \cos^2 \alpha}{\sin^3 \alpha \cos^3 \alpha} = \frac{1}{\sin^3 \alpha \cos^3 \alpha} - \frac{3}{\sin \alpha \cos \alpha} $$
Descomponiendo el resultado esperado $\tan^3 \alpha + \cot^3 \alpha$:
$$ \frac{\sin^3 \alpha}{\cos^3 \alpha} + \frac{\cos^3 \alpha}{\sin^3 \alpha} = \frac{\sin^6 \alpha + \cos^6 \alpha}{\sin^3 \alpha \cos^3 \alpha} = \frac{1 - 3\sin^2 \alpha \cos^2 \alpha}{\sin^3 \alpha \cos^3 \alpha} $$
Ambos lados coinciden.

3. Resultado final:
$$ \boxed{\tan^3 \alpha + \cot^3 \alpha = \tan^3 \alpha + \cot^3 \alpha} $$