$f(x) = \sin (1 - 2x)^³$
Is $\sin(1-2x)^³$ same as $\sin^³(1-2x)$?
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1Short answer: no. – Ethan Bolker Jun 20 '23 at 11:15
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2This form $\sin (1 - 2x)^3$ for $ \sin^3 (1 - 2x)$ should be avoided, we can use as an alternative $(\sin (1 - 2x))^3$. – user Jun 20 '23 at 11:27
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Please refer to my explanation at What does $\operatorname{trig}(x)^n$ mean?. The short answer is that some of us actively avoid $\sin(x)^n$ for being, at best, unnecessarily ambiguous, even as many people consider $\sin(x)^n$ and $\sin^n(x)$ as synonyms. – ryang Jun 21 '23 at 14:03
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You have to be careful with the parentesis. Note that $$\sin^3(1-2x)=\sin (1-2x)\cdot \sin (1-2x)\cdot \sin (1-2x),$$ and $$\sin (1-2x)^3$$ is a little ambigous, it should be written as $$\sin \left((1-2x)^3\right)=\sin \left((1-2x)\cdot (1-2x)\cdot (1-2x)\right),$$ which is clearly not the same.
