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Given -π/6 < θ < -π/12, let α₁ and β₁ be the solutions of the quadratic equation x² - 2x secθ + 1 = 0, and α₂ and β₂ be the solutions of the quadratic equation x² + 2x tanθ - 1 = 0. If α₁ > β₁ and α₂ > β₂, what is the value of α₁ + β₂?

1. 2(secθ - tanθ)
2. 2 secθ
3. -2 tanθ
4. 0

Correct answer: -2 tanθ

Solution

The roots of the quadratic equations are derived using the quadratic formula. Adding α₁ and β₂ involves simplifying the expressions for the roots, which results in -2 tanθ as the sum.

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