Using exact trigonometric ratios, evaluate the value of sec 45° + tan 30° and express the result in simplest surd form.

Introduction / Context:
This question checks your recall of exact trigonometric values for 30° and 45° and your ability to manipulate surds. You must compute sec 45° and tan 30°, then add them and simplify the result into a single compact expression involving square roots, as commonly required in exam style answers.

Given Data / Assumptions:

• Expression: sec 45° + tan 30°.
• Angles are measured in degrees.
• Known exact values: sin 30°, cos 30°, sin 45°, cos 45°.
• We aim for a simplified surd expression.

Concept / Approach:
We use the definitions sec θ = 1 / cos θ and tan θ = sin θ / cos θ. For 45°, cos 45° = √2/2, so sec 45° is its reciprocal. For 30°, sin 30° = 1/2 and cos 30° = √3/2, so tan 30° = sin 30° / cos 30°. After substituting, we add the two resulting terms and then combine them into a single fraction with a common denominator, simplifying any surds that arise.

Step-by-Step Solution:
cos 45° = √2/2, so sec 45° = 1 / (√2/2) = 2/√2 = √2. For 30°, sin 30° = 1/2 and cos 30° = √3/2. Hence tan 30° = (1/2) / (√3/2) = 1/√3. Therefore sec 45° + tan 30° = √2 + 1/√3. To combine into a single fraction, express √2 with denominator √3. Write √2 = √2 · (√3/√3) = √6/√3. So √2 + 1/√3 = (√6/√3) + (1/√3) = (√6 + 1)/√3.

Verification / Alternative check:
Check numerically: √2 ≈ 1.414 and 1/√3 ≈ 0.577. Their sum is about 1.991. Now compute (√6 + 1)/√3: √6 ≈ 2.449, so numerator is roughly 3.449. Divide by √3 ≈ 1.732 to get about 1.991 again. The close match confirms that the simplified form (√6 + 1)/√3 is correct.

Why Other Options Are Wrong:
(1 + √3)/2 evaluates to roughly 1.366, which does not match 1.991. (√3 + 2)/√3 is about (1.732 + 2)/1.732 ≈ 2.155. The choice 5/√3 is about 2.887, which is far too large. The unsimplified form √2 + 1/√3 is equivalent numerically, but the question expects a single fraction. Only (√6 + 1)/√3 is both simplified and equal to the original expression.

Common Pitfalls:
Mixing up tan 30° with tan 60° is a common mistake, as is forgetting to rationalize or combine terms into a single fraction when needed. Some learners miscalculate sec 45° as 1/√2 instead of √2. Carefully using definitions and standard values for special angles helps avoid these errors.

Final Answer:
Thus, sec 45° + tan 30° simplifies exactly to (√6 + 1)/√3.