\[ \text { Find the value of } \frac{16}{\sqrt{3}}\left(\cos 50^{\circ} \cos 10^{\circ} \cos 110^{\circ} \cos 60^{\circ}\right) \]

Given:
– We need to find the value of \(\frac{16}{\sqrt{3}}\left(\cos 50^\circ \cos 10^\circ \cos 110^\circ \cos 60^\circ\right)\).

1. We use the identity \(\cos x \cos(60 – x) \cos(60 + x) = \frac{1}{4} \cos 3x\):

\[ \cos x \cos(60 – x) \cos(60 + x) = \frac{1}{4} \cos 3x \]

2. Applying this identity to \(\cos 50^\circ, \cos 10^\circ, \cos 110^\circ\):

\[ \cos 50^\circ \cos 10^\circ \cos 110^\circ = \frac{1}{4} \cos 150^\circ \]
\[ \cos 150^\circ = -\frac{\sqrt{3}}{2} \]
\[ \cos 50^\circ \cos 10^\circ \cos 110^\circ = -\frac{\sqrt{3}}{8} \]

3. Also, \(\cos 60^\circ = \frac{1}{2}\).

4. Substituting these values into the given expression:

\[ \frac{16}{\sqrt{3}}\left(\cos 50^\circ \cos 10^\circ \cos 110^\circ \cos 60^\circ\right) \]
\[ = \frac{16}{\sqrt{3}} \times \left(-\frac{\sqrt{3}}{8}\right) \times \frac{1}{2} \]
\[ = -1 \]

Conclusion:
The value of \(\frac{16}{\sqrt{3}}\left(\cos 50^\circ \cos 10^\circ \cos 110^\circ \cos 60^\circ\right)\) is \(-1\).