What is \tan \theta equal to ?

For the next two (02) items that follow : p + q \cot \theta = 3\operatorname{cosec} \theta and q - p \cot \theta = 2\operatorname{cosec} \theta

  1. A. \frac{(2q + 3p)}{(3q - 2p)} ✓
  2. B. \frac{(2q - 3p)}{(3q + 2p)}
  3. C. \frac{(3q + 2p)}{(2q - 3p)}
  4. D. \frac{(3q - 2p)}{(2q + 3p)}

Correct Answer: A. \frac{(2q + 3p)}{(3q - 2p)}

Explanation

Multiplying equations by \sin\theta, we get p\sin\theta + q\cos\theta = 3 and q\sin\theta - p\cos\theta = 2. By solving this system of linear equations for \sin\theta and \cos\theta (using elimination or Cramer's rule), we find \sin\theta = \frac{3p+2q}{p^2+q^2} and \cos\theta = \frac{3q-2p}{p^2+q^2}. Therefore, \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3p+2q}{3q-2p} = \frac{2q+3p}{3q-2p}.

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