# Simplify addition of 4 sin()s into one term

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:

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

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