1. **State the problem:** Simplify the expression $\sin^2(5\pi - x)$. 2. **Recall the identity:** The sine function has the property $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$. Also, sine has a periodicity of $2\pi$ and the identity $\sin(\theta) = \sin(\pi - \theta)$. 3. **Use the periodicity of sine:** Since $5\pi = 2\pi \times 2 + \pi$, we can write $$\sin(5\pi - x) = \sin(\pi - x)$$ 4. **Use the sine subtraction identity:** $$\sin(\pi - x) = \sin \pi \cos x - \cos \pi \sin x = 0 \cdot \cos x - (-1) \cdot \sin x = \sin x$$ 5. **Therefore:** $$\sin^2(5\pi - x) = \sin^2 x$$ **Final answer:** $$\boxed{\sin^2(5\pi - x) = \sin^2 x}$$