What is the value of \tan^{2}165^{\circ}+\cot^{2}165^{\circ}?
- A. 7
- B. 14 ✓
- C. 4\sqrt{3}
- D. 8\sqrt{3}
Correct Answer: B. 14
Explanation
We can write \tan 165^{\circ} = \tan(180^{\circ} - 15^{\circ}) = -\tan 15^{\circ} = -(2 - \sqrt{3}) = \sqrt{3} - 2. Similarly, \cot 165^{\circ} = -\cot 15^{\circ} = -(2 + \sqrt{3}). Substituting these in, we get (\sqrt{3} - 2)^2 + (-(2 + \sqrt{3}))^2 = (7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) = 14.