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I needed to find the derivative of $\cos^3(x)$ and find it in a list of given derivatives, but I could not find $-3\sin(x)\cos^2(x)$, so I waited until I did more problems and the only one left was $-3\cos^3(x)\tan(x)$.

If anyone could show me the steps to convert $-3\sin(x)\cos^2(x)$ into $-3\cos^3(x)\tan(x)$, that would be awesome.

Thanks!

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1 Answer 1

Recall that $\tan{x}=\frac{\sin{x}}{\cos{x}}$. So $$ -3\cos^3{x}\tan{x}=-3\cos^3{x}\cdot\frac{\sin{x}}{\cos{x}}=-3\cos^2{x}\sin{x}. $$

So if you want to convert the derivative to that form, you multiply by $\frac{\cos{x}}{\cos{x}}$. Personally, I think the answer you got should have been the one on the list, as I don't see how anyone would arrive to this other form instead without some unnecessary manipulation.

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Perhaps part of the purpose of the problem was to force the students to do this unnecessary manipulation. –  Michael Lugo Jan 30 '11 at 20:00

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