What is \tan \theta equal to ?

For the next two (02) items that follow : \operatorname{cosec} \theta - \sin \theta = p^3 and \sec \theta - \cos \theta = q^3

  1. A. \frac{p}{q}
  2. B. \frac{q}{p} ✓
  3. C. pq
  4. D. \sqrt{pq}

Correct Answer: B. \frac{q}{p}

Explanation

We have p^3 = \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta}, so p = \frac{\cos^{2/3} \theta}{\sin^{1/3} \theta}. Similarly, q^3 = \frac{1 - \cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta}, so q = \frac{\sin^{2/3} \theta}{\cos^{1/3} \theta}. The ratio \frac{q}{p} = \frac{\sin^{2/3}\theta}{\cos^{1/3}\theta} \times \frac{\sin^{1/3}\theta}{\cos^{2/3}\theta} = \frac{\sin \theta}{\cos \theta} = \tan \theta.

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