What is the intuition behind $\cos^2(x)$ being the same as $(\cos x)^2$?
Shouldn't $(\cos x)^2$ be $\cos^2 \cdot x^2$? as $(xy)^2$ is $x^2y^2$? What is the intuition behind it? Please use simple language as I am not a mathmatician.
$\endgroup$ 13 Answers
$\begingroup$No. This confusion likely arises from the extremely unfortunate notation present in trigonometry. Really, the expression should look more like: $$(f(x))^2$$ where $f$ is a function. This does not equal $$f^2(x^2)$$ in general, regardless of how you define the square of a function (which is itself not a particularly natural idea). That is to say, the identity $$(xy)^2=x^2y^2$$ says "squaring distributes over multiplication" - but an expression like $f(x)$ is not multiplication - it is the application of $f$ to $x$ - and squaring doesn't necessarily do anything to that.
Unfortunately, if we let $f$ be the cosine, one usually write $\cos x$ in place of $\cos(x)$ - this looks like multiplication, but it is not - and hence we can't distribute the square. That is to say, we usually take $$(\cos x)^2$$ to be a shorthand for $$(\cos(x))^2$$ which is often itself shortened as $$\cos^2(x).$$
In short: the issue is purely notational. Those things are defined to be equal - that is, they mean the same thing and are not derived from each other by symbolic manipulation, nor are manipulations which look okay necessarily justified given that the symbols aren't interpreted as usual.
$\endgroup$ $\begingroup$The difference is that $\cos x$ is a function. So $\cos^2(x^2)$ means take $x$, square it, then plug it into $\cos$, then square the answer you get for that. But $(\cos x)^2$ is not the same as $\cos x^2$. Notice $(\cos x)^2$ means take $x$, plug it into $\cos$, then square that value. There is no reason to think that if you square $x$ first, then plug it into $\cos$ that you should get the same value.
This is true for any function $f(x)$. If we write $(f(x))^2$, we mean plug $x$ into $f(x)$, then square that value. If $f(x)$ is a 'weird' function (meaning 'most' functions), there is no reason that $f(x)^2=f(x^2)$. However, there are functions where this is true - take $f(x)=1$.
Now why is $(\cos x)^2=\cos^2 x$? This is just notation. If we write $\cos x^2$, without the parentheses, we mean take $x$, square it, then plug it into $\cos x$. But what if we mean take $x$, plug it into $\cos$, then square? We couldn't write $\cos x^2$, because most people would look at that and say $x$ was being squared, not $\cos$. So we have to write $(\cos x)^2$. But having to write parentheses each time becomes tiresome for mathematicians quickly. So instead we say we can put the exponent right after the trig function so we get to write less. Meaning, we can write $\cos^2 x$ instead of $(\cos x)^2$.
Now why is $(\cos x)^2$ not $\cos^2 x^2$? This is because we have plugged $x$ into $\cos$. So what we have is $\cos(x)$ not $\cos * x$. Meaning, we have plugged $x$ into $\cos$ not multiplied $\cos$ and $x$. However, if we have $(y \cos x)^2$, this is indeed $y^2 \cos^2 x$. This is because for any two functions: $(f(x)g(x))^2=f(x)^2 g(x)^2$.
Math can be weird in this sense. Mathematical notation always makes the most sense to those that first thought of defining that way but not necessarily to others. However, it survives only because it is useful or saves time (like in your question). Hopefully, this helped you!
$\endgroup$ 2 $\begingroup$$\cos x$ is simply a shorthand for $\cos(x)$. $\cos^2 x$ is $(\cos x)^2 by definition. There's no mathematical reason why this is true. It's simply a widely used convention to make expressions slightly more concise.
The expression $\cos^2 \cdot x^2$ is not typical notation. However, depending on the context, $\cos^2 \cdot x^2$ could be interpreted as the function $y \mapsto x^2 \cos^2 y$.
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