Solution of the quadratic equation ${x^2} - \left( {\tan \dfrac{a}{2} + \cot \dfrac{a}{2}} \right)x + \tan \frac{a}{2} \times\cot \frac{a}{2} = 0$ is:

The given quadratic equation is x^2 - (tan(a/2) + cot(a/2))x + tan(a/2)*cot(a/2) = 0, where the product of the roots is 1. One of the roots is tan(a/2), which can be written as sin(a/2) divided by cos(a/2). Multiplying the numerator and denominator by 2cos(a/2) gives 2sin(a/2)cos(a/2) divided by 2cos^2(a/2), which simplifies to sin(a) divided by (1 + cos(a)) using the double angle formulas. This result is the reciprocal of (1 + cos(a))/sin(a), so the other root is (1 + cos(a))/sin(a).

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