Discussion
The value of \(\frac{tan75^{o} + tan15^{o}}{tan75^{o} - tan15^{o}}\) is equal to
Answer: E
The value is \(\frac{2}{\sqrt{3}}\).
Use \(\tan 75^\circ = 2 + \sqrt{3}\) and \(\tan 15^\circ = 2 - \sqrt{3}\).
\[ \begin{aligned} &\frac{(2 + \sqrt{3}) + (2 - \sqrt{3})}{(2 + \sqrt{3}) - (2 - \sqrt{3})} \\ &= \frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}} \end{aligned} \]
Equivalently, the expression is \(\frac{\sin 90^\circ}{\sin 60^\circ}\), since \(\tan A \pm \tan B = \frac{\sin(A \pm B)}{\cos A\cos B}\). \(\sqrt{3}\) and \(\frac{1}{\sqrt{3}}\) are tangents of 60° and 30°, not this ratio.
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