<s>[INST] Proving $1-\frac{1}{2}\sin(2x)=\frac{\sin^3x+\cos^3x}{\sin x+\cos x}$ without factoring Is there a way to prove this identity without factoring?
$$1-\frac{1}{2}\sin(2x)=\frac{\sin^3x+\cos^3x}{\sin x+\cos x}$$
 [/INST] Motivated by the fact that $(a^2 + b^2 - ab)(a+b) =a^3 + b^3$, we see LHS = $1 - \frac{1}{2} \sin(2x) = \sin(x)^2 + \cos(x)^2 - \sin(x) \cos(x)$ and we are done.
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