If \sec \theta+\cos \theta=\frac{5}{2}, where 0 \leq \theta \leq 90^\circ, then what is the value of \sin^2 \theta?

Explanation

Let \cos \theta = x. The equation is x + \frac{1}{x} = \frac{5}{2}. Solving this quadratic gives x = \frac{1}{2} or 2. Since \cos \theta \leq 1, \cos \theta = \frac{1}{2} \implies \theta = 60^\circ. Thus, \sin^2 \theta = \sin^2 60^\circ = \frac{3}{4}.

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