If \(\cot A = x + \dfrac{{1}}{{x}}\), find \(\text{{cosec}}^2 A\).
\(x^2 + \dfrac{{1}}{{x^2}} + 3\)
Using the identity \(\text{{cosec}}^2 A = 1 + \cot^2 A\):
\(\cot^2 A = \left(x + \dfrac{{1}}{{x}}\right)^2 = x^2 + 2 + \dfrac{{1}}{{x^2}}\)
\(\text{{cosec}}^2 A = 1 + x^2 + 2 + \dfrac{{1}}{{x^2}} = \mathbf{{x^2 + \dfrac{{1}}{{x^2}} + 3}}\)
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