Prove the identity: $$ \cos^3x+\sin^3x=(\cos x+\sin x)\cdot(1-\sin x\cdot\cos x) $$ I was able to change both sides to $\cos x-\cos x\cdot\sin^{2}x+\sin x-\sin x\cdot\cos^{2}x$, which is kind of long. Is there a shorter way, such as factoring the left side? If so, how can I do it?
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10$\begingroup$ HINT: Have you seen an identity involving $a^3 + b^3$? $\endgroup$– SrivatsanDec 7, 2011 at 0:24
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2$\begingroup$ Maybe you have seen a factorization of $x^3-y^3$? $\endgroup$– André NicolasDec 7, 2011 at 0:26
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1$\begingroup$ Then the phrase you want to search for is "sum of cubes" (or "difference of cubes"). $\endgroup$– BlueDec 7, 2011 at 0:30
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4$\begingroup$ So you don't know $x^3+y^3=(x+y)(x^2-xy+y^2)$? $\endgroup$– J. M. ain't a mathematicianDec 7, 2011 at 0:31
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2$\begingroup$ Alex Yan: I'm not sure exactly if this is what André intended, but $x^3+y^3$ also has the form $x^3-z^3$ with $z=-y$; if you know one factorization you get the other. $\endgroup$– Jonas MeyerDec 7, 2011 at 0:35
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1 Answer
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Observe that $$ \begin{align*} \cos^3x+\sin^3x &= (\cos x+\sin x)(\cos^2x-\sin x \cos x +\sin^2x) \\ &= (\cos x+\sin x)(1-\sin x \cos x ) \end{align*}$$ noting that $\cos^2x+ \sin^2x=1$.