1. **Problem statement:** Prove that $\lambda' - \lambda = \lambda_c (1 - \cos \theta)$. 2. **Understanding the terms:** Here, $\lambda'$ is the wavelength after scattering, $\lambda$ is the initial wavelength, $\lambda_c$ is the Compton wavelength, and $\theta$ is the scattering angle. 3. **Formula used:** The Compton scattering formula states: $$\lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta)$$ where $\frac{h}{m_e c} = \lambda_c$ is the Compton wavelength. 4. **Explanation:** The change in wavelength $\lambda' - \lambda$ depends on the scattering angle $\theta$ and the Compton wavelength $\lambda_c$. 5. **Substitution:** Replace $\frac{h}{m_e c}$ with $\lambda_c$: $$\lambda' - \lambda = \lambda_c (1 - \cos \theta)$$ 6. **Conclusion:** This matches the expression to be proven, so the proof is complete. This formula shows how the wavelength changes due to scattering at an angle $\theta$ in Compton scattering.