For a general role $f$, which have the right to be about anything and also is $cos$ in her case and with $g(x)=x^n$,$$f^n(x) := (f(x))^n = gcirc f(x) = g(f(x))$$and$$f(x^n) := f(underbracexcdot x cdot ldots_n ext times) = f circ g(x) = f(g(x))$$are two various functions.Note for trigonometric functions, $cos^-1$ periodically refers come $arccos$, and sometimes to $sec = frac1cos$, so you should be careful around exponentiating functions.

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One that the first things you might observe is the $(cos small(xsmall))^2geqslant0$ vice versa, $cos(x^2)$ might be equal to a negative number. (Why?)

In blue, a graph of the duty $colorbluecos^2(x)$ and in red a graph of the function $colorredcos(x^2)$.


Some say, a an excellent plot is worth a million words! :-)

There are couple of functions such that $f^2(x)=f(x^2)$: basically the powers of $x$, $f(x)=x^n$: $f^2(x)=(x^n)^2=(x^2)^n=f(x^2)$.

The preeminence is more often $f^2(x) e f(x^2)$. Just an example with$f(x)=x+1$: $(x+1)^2 e x^2+1$.





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Proving: $cos A cdot cos 2A cdot cos 2^2A cdot cos 2^3A ... cos 2^n-1A = frac sin 2^n A 2^n sin A $
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