Evaluate the exact value of the expression cos 45 degrees + 1/3 using standard trigonometric values, simplify it into a single surd fraction, and then choose the correct form from the options.

Introduction / Context:
This question involves combining a standard trigonometric value with a simple rational number. The goal is to compute cos 45 degrees + 1/3 exactly and then express the result as a single fraction involving √2, matching one of the options provided.

Given Data / Assumptions:

• Angle is 45 degrees, a standard special angle.
• cos 45 degrees = √2/2.
• The expression is cos 45 degrees + 1/3.
• We are asked to produce an exact symbolic expression, not a decimal approximation.

Concept / Approach:
First substitute the known value for cos 45 degrees. Then add 1/3 by finding a common denominator between 2 and 3. Combine the terms into a single fraction and simplify, ensuring the final expression matches one of the given options. Avoid unnecessary rationalisation unless needed for comparison.

Step-by-Step Solution:
Use the standard value: cos 45 degrees = √2/2. Write the expression as √2/2 + 1/3. Find a common denominator for 2 and 3. The least common denominator is 6. Rewrite each term with denominator 6: √2/2 = (3√2)/6 and 1/3 = 2/6. Add the numerators: (3√2)/6 + 2/6 = (3√2 + 2)/6.
Verification / Alternative check:
You can approximate √2 ≈ 1.414. Then cos 45 degrees ≈ 0.707 and cos 45 degrees + 1/3 ≈ 0.707 + 0.333 ≈ 1.040. Evaluating (3√2 + 2)/6 numerically gives approximately (3·1.414 + 2)/6 ≈ (4.242 + 2)/6 ≈ 6.242/6 ≈ 1.040, which confirms the match.

Why Other Options Are Wrong:
Option a ( 2 + √3 ) is much larger numerically and does not involve the correct denominator. Option b ( (2√2 + 1)/√2 ) simplifies to something different and does not equal √2/2 + 1/3. Option c ( (3 + √2)/(3√2) ) and option d ( 5/√3 ) represent unrelated combinations of surds and fractions and give different approximate values.

Common Pitfalls:
A frequent error is to misremember cos 45 degrees as 1/√2 and then fail to simplify correctly. While 1/√2 is equivalent to √2/2, mixing forms can cause confusion when adding fractions. Another pitfall is to approximate early and then try to match approximate decimals to exact surd expressions, which is unnecessary if you keep everything symbolic.

Final Answer:
The exact simplified value of cos 45 degrees + 1/3 is (3√2 + 2)/6.