Ta có
$(1 + cot \alpha) \sin^3 \alpha + (1 + \tan \alpha) \cos^3 \alpha = (1 + \dfrac{\cos \alpha}{\sin \alpha}) \sin^3 \alpha + (1 + \dfrac{\sin \alpha}{\cos \alpha}) \cos^3 \alpha$
$= \sin^3 \alpha + \cos \alpha \sin^2 \alpha + \cos^3 \alpha + \sin \alpha \cos^2 \alpha$
$= (\sin^3 \alpha +\cos^3 \alpha) + (\cos \alpha \sin^2 \alpha+ \sin \alpha \cos^2 \alpha)$
$= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha - \sin \alpha \cos \alpha) + \sin \alpha \cos \alpha (\sin \alpha + \cos \alpha)$
$= (\sin \alpha + \cos \alpha)(\sin^2 \alpha + \cos^2 \alpha)$
$= \sin \alpha + \cos \alpha$
