If $\sin{\alpha}=p$, where $|p| \le 1$ then the quadratic equation whose roots are $\tan{\dfrac{\alpha}{2}}$ and $\cot{\dfrac{\alpha}{2}}$ is -

Let the required quadratic equation be x^2 - (sum of roots)x + (product of roots) = 0. The sum of the roots is tan(alpha/2) + cot(alpha/2), which equals (sin^2(alpha/2) + cos^2(alpha/2)) / (sin(alpha/2)cos(alpha/2)), simplifying to 1 / (sin(alpha/2)cos(alpha/2)), or 2/sin(alpha). Since p = sin(alpha), the sum is 2/p. The product of the roots is tan(alpha/2) multiplied by cot(alpha/2), which is 1. Substituting these into the equation gives x^2 - (2/p)x + 1 = 0, and multiplying by p yields px^2 - 2x + p = 0.

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