Apply quotient or reciprocal identities.

\sin^3\alpha (1+\cot \alpha )+\cos^3 \alpha (1+\tan \alpha )

=\sin^3\alpha \Big( 1+ \dfrac{\cos \alpha }{\sin \alpha } \Big) +\cos^3 \alpha \Big( 1+ \dfrac{\sin \alpha }{\cos \alpha }\Big)

=\sin^3\alpha\dfrac{\sin \alpha +\cos\alpha }{\sin \alpha}+\cos^3 \alpha\dfrac{\sin \alpha +\cos\alpha }{\cos \alpha }

=\sin^2\alpha (\sin \alpha +\cos\alpha)+\cos^2 \alpha(\sin \alpha +\cos\alpha)

= (\sin \alpha +\cos\alpha)( \sin^2 \alpha +\cos^2\alpha)

=\sin \alpha +\cos\alpha