Difficulty: Medium
Correct Answer: (2√3 - 1)/√3
Explanation:
Introduction / Context:
This problem checks familiarity with exact trigonometric ratios for standard angles 30° and 45°. It combines these values in a simple algebraic expression involving secant and tangent and asks for an equivalent simplified form in terms of square roots.
Given Data / Assumptions:
Concept / Approach:
We recall that sec 45° is the reciprocal of cos 45°, and tan 30° is sin 30° divided by cos 30°. Substituting these standard values, we simplify the resulting expression using rational operations with surds and then match it to one of the given options.
Step-by-Step Solution:
cos 45° = √2 / 2, so sec 45° = 1 / (√2 / 2) = 2/√2 = √2
Thus √2 · sec 45° = √2 · √2 = 2
sin 30° = 1/2, cos 30° = √3 / 2
Hence tan 30° = (1/2) / (√3 / 2) = 1/√3
Given expression = 2 − 1/√3
Write 2 as 2√3/√3
2 − 1/√3 = (2√3/√3) − 1/√3 = (2√3 − 1)/√3
Verification / Alternative check:
Approximate √3 ≈ 1.732. Then (2√3 − 1)/√3 ≈ (3.464 − 1)/1.732 ≈ 2.464/1.732 ≈ 1.4226. Evaluate the original expression numerically: √2 ≈ 1.414, sec 45° ≈ 1.414 so the product is 2, and tan 30° ≈ 0.577. Then 2 − 0.577 ≈ 1.423 which matches the approximate value, confirming the simplification.
Why Other Options Are Wrong:
(2√3 − 1)/3 has a different denominator and evaluates to a smaller value. (√3 − 1)/√3 and (√3 − 1)/3 correspond to entirely different expressions. (2√3 + 1)/3 would be larger than 1.5 and does not match the calculated value near 1.42.
Common Pitfalls:
Learners sometimes misremember standard values, such as swapping tan 30° and tan 60°, or incorrectly simplifying 2/√2. Another issue is rationalising denominators in the wrong step, which is not necessary here beyond matching the option forms. Always compute primary trig values accurately first.
Final Answer:
Thus, √2 · sec 45° − tan 30° simplifies exactly to (2√3 − 1)/√3.
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