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

Question

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 ?

Answer 1

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!).

Answer 2

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 $x=0$.