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Are there any identities for trigonometric equations of the form:

$$A\sin(x) + B\sin(y) = \cdots$$ $$A\sin(x) + B\cos(y) = \cdots$$ $$A\cos(x) + B\cos(y) = \cdots$$

I can't find any mention of them anywhere, maybe there is a good reason why there aren't identities for these? Thanks!

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I have heard of asin(x) + bcos(x) = Acos(x+alpha) where A is in terms of a and b .Without a relation between x and y we cannot combine because of the different coefficients a and b . had they been 1 it is possible –  Harish Kayarohanam Aug 15 '13 at 18:34
    
Perhaps you could be a little bit more specific in what kind of identity you are expecting. Personally I find your initial form simple enough but perhaps you are bothered by the $+$ which makes you unable to do a certain type of calculation? –  user88595 Aug 15 '13 at 18:43
    
One simple special case is the identities saying $\sin x+\sin y$ $=\text{a single term, which is a product of half-angle functions}$, and similarly for $\cos x+\cos y$ and $\sin x+\cos y$. –  Michael Hardy Aug 15 '13 at 18:45

2 Answers 2

there are no general formula for these expressions.but may exist when $A$ and $B$ are interrelated .

For example consider triangle $ABC$ where $a,b,\text{ and }c $ are the sides of the triangle and $A,B,\text{ and }C$ are the respective angles opposite to $a,b,\text{ and }c $ then $$c = a\cos B + b\cos A $$ here this is because $a,b ,A\text{ and }B$ are interrelated by laws of triangle.

therefore random values of the angles and the coefficients will not satisfy to form general formula.

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\begin{align} A\sin x + B\sin y & = \sqrt{A^2+B^2}\left( \frac{A}{\sqrt{A^2+B^2}}\sin x + \frac{B}{\sqrt{A^2+B^2}}\cos y\right) \\[10pt] & = \sqrt{A^2+B^2} \left( \sin z\sin x+\cos z\cos y \right) \\[10pt] & = \sqrt{A^2+B^2} \frac{\cos(x+z)-\cos(x-z)+\cos(z+y)+\cos(z-y)}{2} \end{align} Maybe in some special cases like $x+y+z=\pi$ or $x=y$ you might want to think about simplifying further beyond that point. And also try similary things with your other two expressions.

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