Without using a calculator, determine the value of \( \frac{4 \sin 15^{\circ} \cos 15^{\circ}}{2 \sin ^{2} 15^{\circ}-1} \)

To determine the value of \[ \frac{4 \sin 15^{\circ} \cos 15^{\circ}}{2 \sin ^{2} 15^{\circ}-1}, \] we first simplify the numerator and the denominator. ### Step 1: Simplify the numerator The numerator can be simplified using the double angle identity for sine: \[ 4 \sin 15^{\circ} \cos 15^{\circ} = 2 \cdot 2 \sin 15^{\circ} \cos 15^{\circ} = 2 \sin(30^{\circ}) = 2 \cdot \frac{1}{2} = 1. \] ### Step 2: Simplify the denominator Next, we simplify the denominator: \[ 2 \sin^2 15^{\circ} - 1. \] Using the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \), we can express \( \sin^2 15^{\circ} \) as follows: \[ \sin^2 15^{\circ} = \frac{1 - \cos(30^{\circ})}{2}. \] Since \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \), \[ \sin^2 15^{\circ} = \frac{1 - \frac{\sqrt{3}}{2}}{2} = \frac{2 - \sqrt{3}}{4}. \] Now, substituting this into the denominator: \[ 2 \sin^2 15^{\circ} = 2 \cdot \frac{2 - \sqrt{3}}{4} = \frac{2 - \sqrt{3}}{2}. \] So, \[ 2 \sin^2 15^{\circ} - 1 = \frac{2 - \sqrt{3}}{2} - 1 = \frac{2 - \sqrt{3}}{2} - \frac{2}{2} = \frac{2 - \sqrt{3} - 2}{2} = \frac{-\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}. \] ### Step 3: Combine the results Now we substitute the simplified numerator and denominator back into the original expression: \[ \frac{4 \sin 15^{\circ} \cos 15^{\circ}}{2 \sin^2 15^{\circ}-1} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}. \] ### Step 4: Rationalizing the denominator To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{3} \): \[ -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}. \] Thus, the value of \[ \frac{4 \sin 15^{\circ} \cos 15^{\circ}}{2 \sin ^{2} 15^{\circ}-1} \] is \[ \boxed{-\frac{2\sqrt{3}}{3}}. \]