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I apologize if this is a duplicate of another question in advance. I did not find any in the search.

Does $\sin(x)\sin(y)=\sin(xy)$? I am wondering because I need to know if $\sin^2(44)=\sin(44)\sin(44)=\sin(44^2)=\sin(1936)$. If I am wrong, is there a way to simplify $\sin(x)\sin(y)$? Thanks in advance.

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up vote 3 down vote accepted

No, $\sin (x) \sin (y) \neq \sin (xy)$ in general. $\sin 2 \approx 0.9093$ and $\sin 4 \approx -0.7568 \neq 0.9093^2$ There is an identity $\sin(x)\sin(y)=\frac 12(\cos(x-y)-\cos(x+y))$ but you may not find that an improvement. If you set $x=y$ in this you get $\sin^2x=\frac 12(1-\cos(2x))$

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