# Is it valid to say that $\cos^3(x^{4/3})=\cos(x^4)$?

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 ?

• No. They are close near $x=0$ though. – user587192 Nov 24 '18 at 3:12
• Had it been $cos(x^{\frac{4}{3}})^{3}$ then you could have – Akash Roy Nov 24 '18 at 3:16
• Notice that $\cos^3(y) \neq \cos(y^3)$, so there's no reason to expect the equation in the question to be true. – littleO Nov 24 '18 at 3:27
• 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. – 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!).

• Although it's pretty much true, I don't think that that's the most helpful approach. – rafa11111 Nov 24 '18 at 3:25
• @rafa11111 I just want to show the simple counter example that just arose in my head :) – Seewoo Lee Nov 24 '18 at 3:26
• Yes, of course! I just wanted to point out that, IMHO, that's not exactly an answer... – 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. 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 $$x=0$$.

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