What is the value of \tan 65^{\circ}+2\tan 45^{\circ}-2\tan 40^{\circ}-\tan 25^{\circ} ?
- A. 0
- B. 1
- C. 2 ✓
- D. 4
Correct Answer: C. 2
Explanation
Rearrange to group complementary angles: (\tan 65^{\circ} - \tan 25^{\circ}) - 2\tan 40^{\circ} + 2(1). Using the identity \tan(90^{\circ}-\theta) - \tan \theta = 2\cot 2\theta, we have \tan 65^{\circ} - \tan 25^{\circ} = \cot 25^{\circ} - \tan 25^{\circ} = 2\cot 50^{\circ}. Since \cot 50^{\circ} = \tan 40^{\circ}, the first part is 2\tan 40^{\circ}. The expression evaluates to 2\tan 40^{\circ} - 2\tan 40^{\circ} + 2 = 2.