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I was trying to figure out how to simplify the above expression.

I'd imagine it is easy to simplify the sine term into a cosine term by adding a phase shift, but then how do you add the two cosine functions together into one cosine function.

Jade
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    If $b\neq d$ and neither $a$ nor $c$ is zero, I think you're out of luck. Take a look at http://www.wolframalpha.com/input/?i=f(x)%3D2+cos(3x)%2B3+sin(4x) for example; there is no way to fit that to a single cosine function. – David K Nov 04 '16 at 06:17
  • I agree with David K. These are nonlinear functions, so not sure what can be accomplished here if $b \neq d$ and $ac \neq 0$. Especially if by simplifying you mean ending up with one term instead of two summands. – avs Nov 04 '16 at 06:19
  • what if we have the condition that b=d? – Jade Nov 04 '16 at 06:20
  • If $b=d$ then you have a slightly simpler version of this: http://math.stackexchange.com/questions/1694278/how-to-write-x-sin-a-y-sin-b-as-single-trigonometric-funtion/1694540#1694540 -- skip the first few paragraphs and start just before the equation $A_3 \sin(u + \delta_3) = A \cos u + B \sin u.$ On the right hand side of that equation, to translate it into your notation, $A=a$, $B=c$, and $u=bx=dx$. The left hand side of that equation will be what you're looking for. – David K Nov 04 '16 at 06:28
  • Thanks for the link! It had what I was looking for! – Jade Nov 06 '16 at 02:45

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