Converting $\sin(A)+\cos(B)$ to a product? Similar to,
$$ \sin(u) + \sin(v) = 2 \sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$$
$$ \cos(u) + \cos(v) = 2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{v-u}{2}\right)$$

Are there combination laws for expressions such as:
$$ \sin(A) + \cos(B)$$

where $A,B$ are some real angles?

From the comment by @Andrei
$$ \sin(A) + \cos(B) =  \sin(A) + \sin( \frac{\pi}{2} -B)= 2 \sin \left( \frac{A-B}{2} + \frac{\pi}{4} \right) \cos \left( \frac{A+B}{2} - \frac{\pi}{4} \right)$$
