If \text{cosec } \theta - \sin \theta = p^{3} and \sec \theta - \cos \theta = q^{3}, then what is the value of \tan \theta?

Explanation

We can write p^3 = \frac{1-\sin^2\theta}{\sin\theta} = \frac{\cos^2\theta}{\sin\theta} and q^3 = \frac{1-\cos^2\theta}{\cos\theta} = \frac{\sin^2\theta}{\cos\theta}. Dividing q^3 by p^3, we get \frac{q^3}{p^3} = \frac{\sin^2\theta / \cos\theta}{\cos^2\theta / \sin\theta} = \frac{\sin^3\theta}{\cos^3\theta} = \tan^3\theta. Taking the cube root gives \tan\theta = \frac{q}{p}.

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