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. \sec 4\alpha
- B. \csc 4\alpha
- C. 2 \sec 4\alpha
- D. 2 \csc 4\alpha ✓
Correct Answer: D. 2 \csc 4\alpha
Explanation
Using p = \frac{\sin \alpha}{\cos \alpha \cos 2\alpha} and q = \frac{1}{\sin 2\alpha}, we get p+q = \frac{2\sin^2 \alpha + \cos 2\alpha}{\sin 2\alpha \cos 2\alpha}. Substituting \cos 2\alpha = 1 - 2\sin^2 \alpha in the numerator gives \frac{1}{\sin 2\alpha \cos 2\alpha} = \frac{2}{\sin 4\alpha} = 2\csc 4\alpha.