You are watching: (cos(x))^2
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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