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4 years ago I have not been in sight for a math book. Today I had a dumb question. I hope you can understand me.

$(\sin(x))^2$ is $\sin(x^2)$ or $\sin^2(x)$

I believe that to learn calculus, the first thing I have to do is accept that I have difficulties in basic areas such as trigonometry.

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The second one. – ՃՃՃ Nov 10 '12 at 4:40
The second one? – Rakisbro Nov 10 '12 at 4:43
Yes.${}{}{}{}{}$ – ՃՃՃ Nov 10 '12 at 4:45
up vote 4 down vote accepted

$(f(x))^2$ is often written as $f^2(x)$ and in general, $(f(x))^2\ne f(x^2)$

Example, $(\sin x)^2=\sin^2x, (\cosh x)^2=\cosh^2x, (\log x)^2=\log^2x$

Of course, in many functions like Algebraic/Polynomial ones, $f(x)=ax^2+bx+c$ this notation does not make sense.

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You answer is all. thanks – Rakisbro Nov 10 '12 at 4:45
@Rakisbro, welcome anytime. – lab bhattacharjee Nov 10 '12 at 4:46
This notation isn't generally understood to have the same meaning unless the function is trigonometric. – Dan Brumleve Nov 10 '12 at 6:51
@DanBrumleve, is that the reason for down-voting? It's too obvious that we can't use this notation for Algebraic/Polynomial/Exponential/Logarithmic Function unlike for Trigonometric/Hyperbolic. – lab bhattacharjee Nov 10 '12 at 8:41
Yes. Why is it obvious that $\tan^2{x}$ should mean $(\tan{x})^2$ but $\log^2{x}$ should mean something else? I explained my opinion in my own answer. – Dan Brumleve Nov 10 '12 at 8:46

With trigonometric functions, such as $y= \tan(x)$, it is standard to write $$y^2=\tan^2(x)$$

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As Linear Man says, this is a common notation for trigonometric functions. However, in a more general context, $f^2(x)$ may mean $f(f(x))$ rather than $(f(x))^2$ as lab bhattacharjee claims. I think it is a poor notation for any other meaning even when limited to trigonometric functions and I prefer to write $(\sin{x})^2$ or $\sin{(x)}^2$ when I mean that, and I am quite comfortable with $\sin^2{(x)}$ or even $\sin^2{x}$ as a notation for $\sin{(\sin{(x)})}$ or with equivalent unambiguity $\sin{\sin{x}}$.

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