If p=\frac{1}{\text{cosec } \theta+\cot \theta} and q=\text{cosec } \theta, then what is p^{2}-2pq equal to?

Explanation

Using identity \text{cosec}^2\theta - \cot^2\theta = 1, p = \text{cosec } \theta - \cot \theta. Then p^2 - 2pq = p(p-2q) = (\text{cosec } \theta - \cot \theta)(-\text{cosec } \theta - \cot \theta) = -(\text{cosec}^2\theta - \cot^2\theta) = -1.

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