Discussion
\(\frac{3\tan 15^{\circ} - \tan^{3}15^{\circ}}{1 - 3\tan^{2}15^{\circ}}\)
Answer: E
The value is 1.
The expression is the triple-angle formula for tangent:
\[ \begin{aligned} \tan 3\theta \\ &= \frac{3\tan\theta - \tan^{3}\theta}{1 - 3\tan^{2}\theta} \end{aligned} \]
With \(\theta = 15^{\circ}\) it equals \(\tan 45^{\circ} = 1\). No numerical value of \(\tan 15^{\circ}\) is needed; recognising the pattern is the whole question. The other options do not correspond to \(\tan 45^{\circ}\).
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