Why is $\cos^2(\arccos(x)) = x^2$ instead of $\cos(x)$? Since we have $cos^2$, shouldn't we be left with one $cos$ after $\cos(\arccos(x)) = x$?
I would greatly appreciate it if someone could please clarify this concept.
Why is $\cos^2(\arccos(x)) = x^2$ instead of $\cos(x)$?
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algebra-precalculus
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0It depends upon whether $\cos^2 (a)$ means $(\cos (a))^2$ or whether it means $\cos(\cos a)$. Different books use different conventions. If the former $(\cos (\arccos x))^2 = x^2$ of course and $\cos(\cos(\arccos a)) = \cos a$ of course. Like you, I far prefer it to mean $ \cos(\cos (a))$ and took it to mean that at first reading. But clearly this text meant it to be $(\cos (a))^2$. – 2017-02-27
1 Answers
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I'm not sure you understand the notation. $\cos^2(x) = \cos(x)^2=\cos(x)\cos(x)$, not $\cos(\cos(x))$.
Unfortunately the notation $f^n(x)$ is often used to represent functional iteration, which seems to be the cause of your confusion; for example, $f^3(x)$ could mean $f(f(f(x)))$.
Why not just write the exponent after the parentheses to avoid ambiguity? Historical precedence. Understanding what is meant is usually contextual. A proof is thus as follows: $$\cos^2(\arccos(x))=\cos(\arccos(x))\cos(\arccos(x)) = x\cdot x = x^2$$