If \(2\cos^2 x - 1 = \frac{1}{2}\) for \(0° < x < 90°\), then the value of \(\sec 2x + \operatorname{cosec} x + \cot^2 x\) is
7
Recall the double-angle identity \(2\cos^2 x - 1 = \cos 2x\).
So \(\cos 2x = \frac{1}{2}\), which means \(2x = 60°\) and therefore \(x = 30°\) (valid since \(0° < x < 90°\)).
Evaluate each term. \(\sec 2x = \sec 60° = 2\).
\(\operatorname{cosec} x = \operatorname{cosec} 30° = 2\).
\(\cot^2 x = \cot^2 30° = (\sqrt{3})^2 = 3\).
Add them: \(2 + 2 + 3 = 7\).
Hence, the value of \(\sec 2x + \operatorname{cosec} x + \cot^2 x\) is 7.
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