1. **State the problem:** Simplify the expression $$\frac{\sin(x+y)+\sin(x-y)}{\cos(x+y)+\cos(x-y)}$$. 2. **Recall sum-to-product formulas:** - $$\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$$ - $$\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$$ 3. **Apply the formulas to numerator:** $$\sin(x+y) + \sin(x-y) = 2 \sin \left( \frac{(x+y)+(x-y)}{2} \right) \cos \left( \frac{(x+y)-(x-y)}{2} \right)$$ Simplify inside the arguments: $$= 2 \sin \left( \frac{2x}{2} \right) \cos \left( \frac{2y}{2} \right) = 2 \sin x \cos y$$ 4. **Apply the formulas to denominator:** $$\cos(x+y) + \cos(x-y) = 2 \cos \left( \frac{(x+y)+(x-y)}{2} \right) \cos \left( \frac{(x+y)-(x-y)}{2} \right)$$ Simplify inside the arguments: $$= 2 \cos \left( \frac{2x}{2} \right) \cos \left( \frac{2y}{2} \right) = 2 \cos x \cos y$$ 5. **Rewrite the original expression:** $$\frac{2 \sin x \cos y}{2 \cos x \cos y}$$ 6. **Cancel common factors:** $$= \frac{\cancel{2} \sin x \cancel{\cos y}}{\cancel{2} \cos x \cancel{\cos y}} = \frac{\sin x}{\cos x}$$ 7. **Recognize the simplified form:** $$\frac{\sin x}{\cos x} = \tan x$$ **Final answer:** $$\tan x$$