What is (p/q) equal to?
Consider the following for the three (03) items that follow : Let p=\tan 2\alpha-\tan \alpha and q=\cot \alpha-\cot 2\alpha.
- A. -\tan \alpha \cdot \tan 2\alpha
- B. -\cot \alpha \cdot \cot 2\alpha
- C. \tan \alpha \cdot \tan 2\alpha ✓
- D. \cot \alpha \cdot \cot 2\alpha
Correct Answer: C. \tan \alpha \cdot \tan 2\alpha
Explanation
p = \frac{\sin(2\alpha-\alpha)}{\cos 2\alpha \cos \alpha} = \frac{\sin \alpha}{\cos 2\alpha \cos \alpha} and q = \frac{\cos \alpha}{\sin \alpha} - \frac{\cos 2\alpha}{\sin 2\alpha} = \frac{\sin(2\alpha-\alpha)}{\sin \alpha \sin 2\alpha} = \frac{\sin \alpha}{\sin \alpha \sin 2\alpha}. Thus p/q = \frac{\sin \alpha \sin 2\alpha}{\cos \alpha \cos 2\alpha} = \tan \alpha \tan 2\alpha.