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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$\begingroup$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.
$\endgroup$ $\begingroup$Since
$$ A \cos(a+b) = A \cos(a) \cos(b) - A \sin(a) \sin(b) \ \ \ \ \ \ (1) \\ B \cos(a-b) = B \cos(a) \cos(b) + B \sin(a) \sin(b) \ \ \ \ \ \ (2) $$
(1) + (2) gives
$$ A \cos(x) + B \cos(y) = (A+B) \cos(\frac{x+y}{2}) \cos(\frac{x-y}{2}) + (B-A) \sin(\frac{x+y}{2}) \sin(\frac{x-y}{2}) $$
where
$$ x = a + b \\ y = a - b $$
substitute
$$ Q = (A+B) \cos(\frac{x-y}{2}) \\ R = (B-A) \sin(\frac{x-y}{2}) \\ P = \frac{x+y}{2} $$
then
$$ A \cos(x) + B \cos(y) = Q \cos P + R \sin P = \sqrt{Q^2+R^2} \cos(P-\phi) $$
where
$$ \sin \phi = \frac{R}{\sqrt{Q^2+R^2}} \\ \cos \phi = \frac{Q}{\sqrt{Q^2+R^2}} $$
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