Consider the following numbers: I. \tan 22.5^{\circ} II. \cot 22.5^{\circ} III. \tan 22.5^{\circ} - \cot 22.5^{\circ} How many of the above are irrational numbers?
- A. None
- B. Only one
- C. Only two ✓
- D. All three
Correct Answer: C. Only two
Explanation
We know that \tan 22.5^{\circ} = \sqrt{2}-1 and \cot 22.5^{\circ} = \sqrt{2}+1, both of which are irrational. Their difference is (\sqrt{2}-1) - (\sqrt{2}+1) = -2, which is a rational number. Thus, exactly two of the given numbers are irrational.