As the title says, is it valid to insert the power of the cosine to its angle? Edit : Is it valid when x is very small ?

  • 1
    $\begingroup$ No. They are close near $x=0$ though. $\endgroup$ – user587192 Nov 24 '18 at 3:12
  • $\begingroup$ Had it been $cos(x^{\frac{4}{3}})^{3}$ then you could have $\endgroup$ – Akash Roy Nov 24 '18 at 3:16
  • $\begingroup$ Notice that $\cos^3(y) \neq \cos(y^3)$, so there's no reason to expect the equation in the question to be true. $\endgroup$ – littleO Nov 24 '18 at 3:27
  • $\begingroup$ When $x\to 0$, $$\cos^3(x^{4/3})=1-\frac{3}{2}x^{8/3} + o(x^4)$$ while $$\cos(x^4) = 1-\frac{1}{2}x^8 + o(x^8)$$ so the first non-constant terms do not even closely match up. $\endgroup$ – Clement C. Nov 24 '18 at 3:43

For $x = \pi^{3/4}$, $\cos^{3}(x^{4/3}) = \cos^{3}(\pi) = -1$. However, $\cos(\pi^{4})\neq -1$, since $\pi^{4}$ can't be a rational multiple of $\pi$ (since $\pi$ is a transcendental number!).

  • $\begingroup$ Although it's pretty much true, I don't think that that's the most helpful approach. $\endgroup$ – rafa11111 Nov 24 '18 at 3:25
  • $\begingroup$ @rafa11111 I just want to show the simple counter example that just arose in my head :) $\endgroup$ – Seewoo Lee Nov 24 '18 at 3:26
  • $\begingroup$ Yes, of course! I just wanted to point out that, IMHO, that's not exactly an answer... $\endgroup$ – rafa11111 Nov 24 '18 at 3:29

Although $$ (x^{4/3})^3 = x^4, $$ one does not have $$ [f(x^{4/3})]^3=f(x^4) $$ in general.

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For your added question "Is it valid when $x$ is very small?":

I assume that you mean when $|x|$ is very small.

No. If these two functions are identical near $x=0$, then they must have the same Taylor expansion. But it is not difficult to see by comparing a few terms that they don't have the same Taylor expansion near $x=0$.

  • $\begingroup$ So this means that this is valid when x is very small ? $\endgroup$ – John adams Nov 24 '18 at 3:21
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    $\begingroup$ @Johnadams: "close" does not necessarily mean they are "identical". But one can tell that they are identical at $x=0$, of course. $\endgroup$ – user587192 Nov 24 '18 at 3:41

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