Is $\cos(x^2)$ the same as $\cos^2(x)$?
I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
$\endgroup$ 25 Answers
$\begingroup$For a general function $f$, which can be about anything and is $\cos$ in your case and with $g(x)=x^n$,
$$f^n(x) := (f(x))^n = g\circ f(x) = g(f(x))$$
and
$$f(x^n) := f(\underbrace{x\cdot x \cdot \ldots}_{n \text{ times}}) = f \circ g(x) = f(g(x))$$
are two different functions.
Note for trigonometric functions, $\cos^{-1}$ sometimes refers to $\arccos$, and sometimes to $\sec = \frac1{\cos}$, so you should be careful about exponentiating functions.
One of the first things you may observe is that $(\cos {\small{(}}x{\small{)}})^2\geqslant0$ whereas $\cos(x^2)$ may be equal to a negative number. (Why?)
In blue, a graph of the function $\color{blue}{\cos^2(x)}$ and in red a graph of the function $\color{red}{\cos(x^2)}$.
Some say, a good plot is worth a million words! :-)
$\endgroup$ 0 $\begingroup$There are few functions such that $f^2(x)=f(x^2)$: essentially the powers of $x$, $f(x)=x^n$: $f^2(x)=(x^n)^2=(x^2)^n=f(x^2)$.
The rule is more often $f^2(x)\ne f(x^2)$. Just an example with$f(x)=x+1$: $(x+1)^2\ne x^2+1$.
$\endgroup$ $\begingroup$No $\cos^2(x)$ means $(\cos x)^2$. This is not the same as $\cos(x^2)$.
$\endgroup$ 2 $\begingroup$No. $\cos x^2=\cos(x^2)=\cos(x\times x)$ while $\cos^2(x)=\cos(x)\times \cos(x)=(\cos(x))^2$.
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