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نعبر عن \(75^\circ\) كمجموع \(45^\circ + 30^\circ\):
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\[
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\tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}
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نعوض بالقيم المعروفة \(\tan 45^\circ = 1\) و \(\tan 30^\circ = \frac{1}{\sqrt{3}}\):
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\tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \times \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}
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نُنا)">
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