$\frac{\cos ^{3} \theta+\sin ^{3} \theta}{\cos \theta+\sin \theta}+\frac{\cos ^{3} \theta-\sin ^{3} \theta}{\cos \theta-\sin \theta}=2$
$\mathrm{LHS}=\frac{\cos ^{3} \theta+\sin ^{3} \theta}{\cos \theta+\sin \theta}+\frac{\cos ^{3} \theta-\sin ^{3} \theta}{\cos \theta-\sin \theta}$
$=\frac{(\cos \theta+\sin \theta)\left(\cos ^{2} \theta-\cos \theta \sin \theta+\sin ^{2} \theta\right)}{(\cos \theta+\sin \theta)}+\frac{(\cos \theta-\sin \theta)\left(\cos ^{2} \theta+\cos \theta \sin \theta+\sin ^{2} \theta\right)}{(\cos \theta-\sin \theta)}$
$=\left(\cos ^{2} \theta+\sin ^{2} \theta-\cos \theta \sin \theta\right)+\left(\cos ^{2} \theta+\sin ^{2} \theta+\cos \theta \sin \theta\right)$
$=(1-\cos \theta \sin \theta)+(1+\cos \theta \sin \theta)$
$=2$
$=$ RHS
Hence, LHS= RHS