<s>[INST] Trigonometric Identities Like $A \sin(x) + B \cos(y) = \cdots$ 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!
 [/INST] 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
$$
\sin \phi = \frac{R}{\sqrt{Q^2+R^2}} \\
\cos \phi = \frac{Q}{\sqrt{Q^2+R^2}}
$$
</s>