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1. If \(\frac{cos^2 \ \theta}{sin \ \theta} = p\) and \(\frac{sin^2 \ \theta}{cos \ \theta} = q\), then prove that \(p^2 q^2 (p^2 + q^2 + 3) = 1\).
2. Prove that \(\frac{(1 + cot \ A + tan \ A)(sin \ A - cos \ A)}{sec^3 \ A - cosec^3 \ A} = sin^2 \ A cos^2 \ A\)
3. Prove that \(\frac{\sin \ A}{\sec \ A + \tan \ A - 1} + \frac{\cos \ A}{\text{cosec}\: A + \cot \ A - 1} = 1\)
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