Evaluate: [sin(30°) + tan(45°) - csc(60°)] / [sec(30°) + cos(60°) + cot(45°)].
Answer and explanation
Correct answer: A. (43 - 24√3) / 11
- Step 1: Numerator: 1/2 + 1 - 2/√3 = 3/2 - 2/√3 = (3√3 - 4) / (2√3).
- Step 2: Denominator: 2/√3 + 1/2 + 1 = 2/√3 + 3/2 = (4 + 3√3) / (2√3).
- Step 3: Fraction = (3√3 - 4) / (3√3 + 4).
- Step 4: Rationalize: (3√3 - 4)^2 / (27 - 16) = (27 + 16 - 24√3) / 11 = (43 - 24√3) / 11.
Why not the others: Other options result from confusing reciprocal trigonometric ratios, applying incorrect complementary angle identities, or algebraic slips in squaring ratios.
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