1. **State the problem:** Solve the equation $$\cos\left(\frac{x}{2}-1\right) = \cos^2\left(1 - \frac{x}{2}\right).$$ 2. **Rewrite the equation:** Let $$y = \frac{x}{2} - 1.$$ Then the equation becomes $$\cos(y) = \cos^2(-y) = \cos^2(y)$$ because cosine is an even function. 3. **Rewrite the equation with variable $$y$$:** $$\cos(y) = \cos^2(y).$$ 4. **Rearrange:** $$\cos^2(y) - \cos(y) = 0$$ 5. **Factor:** $$\cos(y) \left( \cos(y) - 1 \right) = 0.$$ This gives two cases: - Case 1: $$\cos(y) = 0$$ - Case 2: $$\cos(y) - 1 = 0 \Rightarrow \cos(y) = 1$$ 6. **Solve Case 1:** $$\cos(y) = 0$$ implies $$y = \frac{\pi}{2} + k\pi$$ for any integer $$k$$. 7. **Solve Case 2:** $$\cos(y) = 1$$ implies $$y = 2k\pi$$ for any integer $$k$$. 8. **Recall substitution:** $$y = \frac{x}{2} - 1$$, so - From Case 1: $$\frac{x}{2} - 1 = \frac{\pi}{2} + k\pi \Rightarrow x = 2 + \pi + 2k\pi$$ - From Case 2: $$\frac{x}{2} - 1 = 2k\pi \Rightarrow x = 2 + 4k\pi$$ **Final solution:** $$x = 2 + \pi + 2k\pi \quad\text{or}\quad x = 2 + 4k\pi$$ for any integer $$k$$.