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How can we compute the sum $$ \sin(f_1) + \sin(f_2) $$ I know it is $$ 2\sin\left(\frac{f_2 + f_1}{2}\right) \cos\left(\frac{f_2 - f_1}{2}\right) $$ but how can it be derived with elementary trigonometric identites?

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The sine angle addition rule reads:

$$\sin(u\pm v)=\sin(u)\cos(v)\pm\sin(v)\cos(u)$$

You can prove your identity by rewriting $f_1=\frac{f_2+f_1}{2}-\frac{f_2-f_1}{2}$ and similarily for $f_2$ and applying the sine addition formula twice.

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$$f_1= \frac{f_1+f_2}{2}+\frac{f_1-f_2}{2}$$ $$f_2= \frac{f_1+f_2}{2}-\frac{f_1-f_2}{2}$$

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$$\sin(f_1) + \sin(f_2)=\sin \cfrac {2f_1}{2}+ \sin\cfrac {2f_2}{2}$$

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