Answer
$\dfrac{\sin^{3}x+\cos^{3}x}{\sin x+\cos x}=1-\sin x\cos x$
Work Step by Step
$\dfrac{\sin^{3}x+\cos^{3}x}{\sin x+\cos x}=1-\sin x\cos x$
Factor the numerator and simplify:
$\dfrac{(\sin x+\cos x)(\sin^{2}x-\sin x\cos x+\cos^{2}x)}{\sin x+\cos x}=1-\sin x\cos x$
$\sin^{2}x-\sin x\cos x+\cos^{2}x=1-\sin x\cos x$
Since $\sin^{2}x+\cos^{2}x=1$, the identity is proved:
$1-\sin x\cos x=1-\sin x\cos x$
