1. **Stating the problem:** Given that $A + B = \frac{\pi}{4}$, we want to simplify the expression $\frac{\cos B - \sin B}{\cos B + \sin B}$. 2. **Rewrite the expression:** The expression is $\frac{\cos B - \sin B}{\cos B + \sin B}$. 3. **Use the identity for tangent of a sum:** Since $A + B = \frac{\pi}{4}$, we have $B = \frac{\pi}{4} - A$. 4. **Express cosine and sine of $B$ in terms of $A$: $$\cos B = \cos\left(\frac{\pi}{4} - A\right) = \cos \frac{\pi}{4} \cos A + \sin \frac{\pi}{4} \sin A = \frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A$$ $$\sin B = \sin\left(\frac{\pi}{4} - A\right) = \sin \frac{\pi}{4} \cos A - \cos \frac{\pi}{4} \sin A = \frac{\sqrt{2}}{2} \cos A - \frac{\sqrt{2}}{2} \sin A$$ 5. **Substitute into the expression:** $$\frac{\cos B - \sin B}{\cos B + \sin B} = \frac{\left(\frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A\right) - \left(\frac{\sqrt{2}}{2} \cos A - \frac{\sqrt{2}}{2} \sin A\right)}{\left(\frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A\right) + \left(\frac{\sqrt{2}}{2} \cos A - \frac{\sqrt{2}}{2} \sin A\right)}$$ 6. **Simplify numerator:** $$\frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A - \frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A = \frac{\sqrt{2}}{2} \sin A + \frac{\sqrt{2}}{2} \sin A = \sqrt{2} \sin A$$ 7. **Simplify denominator:** $$\frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \sin A + \frac{\sqrt{2}}{2} \cos A - \frac{\sqrt{2}}{2} \sin A = \frac{\sqrt{2}}{2} \cos A + \frac{\sqrt{2}}{2} \cos A = \sqrt{2} \cos A$$ 8. **Final simplified expression:** $$\frac{\sqrt{2} \sin A}{\sqrt{2} \cos A} = \frac{\sin A}{\cos A} = \tan A$$ **Answer:** $$\frac{\cos B - \sin B}{\cos B + \sin B} = \tan A$$