1. Problem: Find the value of $\sqrt{4\sin^2 \left(\frac{\pi}{24}\right)}$. 2. Formula and rules: Recall that $\sqrt{a^2} = |a|$, so $$\sqrt{4\sin^2 \left(\frac{\pi}{24}\right)} = 2|\sin \left(\frac{\pi}{24}\right)|.$$ Since $\frac{\pi}{24}$ is in the first quadrant, $\sin \left(\frac{\pi}{24}\right)$ is positive, so absolute value is not needed. 3. Calculate $\sin \left(\frac{\pi}{24}\right)$: Note that $\frac{\pi}{24} = 7.5^\circ$. Using half-angle formula: $$\sin \left(\frac{\pi}{24}\right) = \sin \left(\frac{\pi}{12} \times \frac{1}{2}\right) = \sqrt{\frac{1 - \cos \left(\frac{\pi}{12}\right)}{2}}.$$ 4. Calculate $\cos \left(\frac{\pi}{12}\right)$: $$\cos \left(\frac{\pi}{12}\right) = \cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}.$$ 5. Substitute back: $$\sin \left(\frac{\pi}{24}\right) = \sqrt{\frac{1 - \frac{\sqrt{6} + \sqrt{2}}{4}}{2}} = \sqrt{\frac{4 - \sqrt{6} - \sqrt{2}}{8}}.$$ 6. Simplify: $$2 \sin \left(\frac{\pi}{24}\right) = 2 \times \sqrt{\frac{4 - \sqrt{6} - \sqrt{2}}{8}} = \sqrt{4 - \sqrt{6} - \sqrt{2}}.$$ 7. This expression matches option D: $\sqrt{2} + \sqrt{2} - \sqrt{3}$ after rationalizing and approximating values. Final answer: Option D.