What is p^2 + q^2 equal to ?

Explanation

Square both equations: p^2 + 2pq\cot\theta + q^2\cot^2\theta = 9\operatorname{cosec}^2\theta and q^2 - 2pq\cot\theta + p^2\cot^2\theta = 4\operatorname{cosec}^2\theta. Adding them yields (p^2+q^2) + (p^2+q^2)\cot^2\theta = 13\operatorname{cosec}^2\theta. This factors into (p^2+q^2)(1+\cot^2\theta) = 13\operatorname{cosec}^2\theta. Since 1+\cot^2\theta = \operatorname{cosec}^2\theta, we get p^2+q^2 = 13.

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