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Is it possible to turn $\sin(a)+\sin(b)+\sin(c)+\sin(d)$ into one term, such that there is no addition or subtraction? I've tried by using $$ \sin(a) + \sin(b) = 2 \sin \left(\frac{a+b}{2} \right)\cos \left(\frac{a-b}{2} \right) $$ but that gets it down to 2 terms. I can see it may not be "simpler" but I'm hoping it will make it easier to differentiate to find roots.

Would it work in a similar way for cos?

Edit: a through d are functions of x and y, for example:

$$a=\sqrt{(1+x)^2+(2+y)^2}, b=\sqrt{(1+x)^2+(2-y)^2}, c=\sqrt{(1-x)^2+(2+y)^2}, d=\sqrt{(1-x)^2+(2-y)^2} $$

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  • $\begingroup$ What are you trying to do with the simplified expression? Differentiate it? $\endgroup$
    – John_dydx
    Aug 15, 2015 at 13:10
  • $\begingroup$ Yes, to find roots by numerical methods, and differentiate more than once to find maxima/minima. $\endgroup$
    – User2468
    Aug 15, 2015 at 13:12
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    $\begingroup$ Give an example of a function, for which you want to have roots or extrema ? $\endgroup$
    – Peter
    Aug 15, 2015 at 13:16
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    $\begingroup$ what else is given? $\endgroup$
    – Dr. Sonnhard Graubner
    Aug 15, 2015 at 13:17
  • $\begingroup$ If the derivates get too complicated, you can also use regula falsi (if the functions have only one variable). $\endgroup$
    – Peter
    Aug 15, 2015 at 13:22

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