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Let $x, y \in \Bbb R$. Show that there is $\theta \in (0, 1)$ such that $$\sin(x + y) = x + y − \frac{1}{2}(x^2 + 2xy + y^2 ) \sin(\theta(x + y))$$

It seems like I need to somehow use Taylor's theorem to do this proof. I tried to use trig identity to write $\sin(x+y)=\sin x \cos y + \cos x \sin y$ and then do Taylor expansion.

But I'm confused by where the function is centered and how many degrees should I take?

Thank you.

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Try using the Taylor expansion for $\sin z$, and then substitute $z=x+y$.

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  • $\begingroup$ Thanks. I just take the first two terms of Maclaurin series of sinz with reminder in Lagrange form, then the statement follows. $\endgroup$ – C. L Nov 14 '15 at 22:20

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