If cos 45° − sec 30° = x, then what is the exact simplified value of x in radical form?

Introduction / Context:
This question tests your knowledge of exact trigonometric values for special angles and your ability to combine them into a single simplified radical expression. It involves both cosine and secant functions and requires careful handling of surds and common denominators.

Given Data / Assumptions:

• x = cos 45° − sec 30°.
• We must express x in exact radical form.
• Standard values for sine and cosine of 30° and 45° are known.

Concept / Approach:
Main ideas:

• Recall cos 45° and cos 30° in radical form.
• Use the definition sec θ = 1 / cos θ.
• Convert both terms to a common denominator for simplification.
• Express the final result in the simplest radical form given in the options.

Step-by-Step Solution:
Recall cos 45° = √2 / 2. Also cos 30° = √3 / 2, so sec 30° = 1 / cos 30° = 2 / √3. Thus x = cos 45° − sec 30° = √2 / 2 − 2 / √3. To combine these, use common denominator 2√3. Rewrite √2 / 2 as (√2 * √3) / (2√3) = √6 / (2√3). Rewrite 2 / √3 as 4 / (2√3). Therefore x = √6 / (2√3) − 4 / (2√3). Combine over the common denominator: x = (√6 − 4) / (2√3).

Verification / Alternative check:
Approximate numerically: cos 45° ≈ 0.707 and sec 30° ≈ 1.155. So x ≈ 0.707 − 1.155 ≈ −0.448. Now evaluate (√6 − 4) / (2√3) using √6 ≈ 2.449 and √3 ≈ 1.732. Numerator ≈ 2.449 − 4 = −1.551. Denominator ≈ 2 * 1.732 = 3.464. The quotient ≈ −1.551 / 3.464 ≈ −0.448, matching the earlier estimate.

Why Other Options Are Wrong:
Option b: (4 − √6) / (2√3) is the negative of the correct numerator order and would give a positive value, not matching cos 45° − sec 30°.
Option c and option d: These have a plus sign in the numerator, suggesting addition rather than subtraction of the two trigonometric values.
Option e: √3 − √2 is not over a common denominator and evaluates to a different numeric value than the expression involving cosine and secant.

Common Pitfalls:
Students often confuse sec 30° with cos 30° or fail to rationalise or match denominators correctly. Another common issue is reversing the order of subtraction, which changes the sign of the result. Keeping track of exact values and sign order ensures the correct radical form.

Final Answer:
The exact simplified value of x is (√6 − 4) / (2√3).