Given the quadratic expression \( 2x^2 - 4x + 5 \), complete the square and then factor the resulting expression. Next, calculate the determinant of the 3x3 matrix formed by the coefficients of the completed square expression and solve the resulting polynomial equation. Finally, use trigonometric identities to simplify the expression \(\sin^2(\theta) + \cos^2(\theta)\). Let's think step by step and output the final answer within \boxed{}.
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