Using standard values: $\sin 60° = \dfrac{\sqrt{3}}{2}$, $\tan 30° = \dfrac{1}{\sqrt{3}}$, $\cos 45° = \dfrac{1}{\sqrt{2}}$
$$\frac{\sin^3 60° - \tan 30°}{\cos^2 45°} = \frac{\left(\dfrac{\sqrt{3}}{2}\right)^3 - \dfrac{1}{\sqrt{3}}}{\left(\dfrac{1}{\sqrt{2}}\right)^2}$$
$$= \frac{\dfrac{3\sqrt{3}}{8} - \dfrac{1}{\sqrt{3}}}{\dfrac{1}{2}}$$
$$= \frac{\dfrac{3\sqrt{3}}{8} - \dfrac{\sqrt{3}}{3}}{\dfrac{1}{2}} = \frac{\dfrac{9\sqrt{3} - 8\sqrt{3}}{24}}{\dfrac{1}{2}} = \frac{\dfrac{\sqrt{3}}{24}}{\dfrac{1}{2}} = \frac{\sqrt{3}}{24} \times 2 = \frac{\sqrt{3}}{12}$$
Source: Chapter 8, Section 8.3 (Table 8.1)
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