Evaluate the exact value of tan 45° + (4/√3) × sec 60° by using standard trigonometric values for special angles and simplifying the final surd expression.

Introduction / Context:
This trigonometric simplification question combines standard values of tan and sec for special angles and requires you to manage surds carefully. The expression involves tan 45 degrees and sec 60 degrees multiplied by a fraction in terms of √3. By recalling the exact values for these special angles and simplifying step by step, you can express the final result as a single surd fraction that matches one of the given options.

Given Data / Assumptions:

• tan 45 degrees is a standard value.
• sec 60 degrees is the reciprocal of cos 60 degrees.
• cos 60 degrees = 1/2, sin 60 degrees = √3/2.
• The expression to evaluate is tan 45 degrees + (4/√3) × sec 60 degrees.
• All calculations are based on exact surd values, not decimal approximations.

Concept / Approach:
Begin by substituting the known exact values for tan 45 degrees and sec 60 degrees. Then multiply the fraction (4/√3) by sec 60 degrees. After this multiplication, you will have a sum of 1 and a surd fraction. Finally, express everything over a common denominator √3 and combine the terms to obtain a single surd fraction that can be compared with the options.

Step-by-Step Solution:
Recall tan 45 degrees = 1. Recall cos 60 degrees = 1/2, so sec 60 degrees = 1 / cos 60 degrees = 2. Substitute into the expression: tan 45 degrees + (4/√3) × sec 60 degrees = 1 + (4/√3) × 2. Compute the product: (4/√3) × 2 = 8/√3. So the expression becomes 1 + 8/√3. Write 1 as √3/√3 to get a common denominator: 1 = √3/√3. Add the fractions: √3/√3 + 8/√3 = (√3 + 8)/√3.
Verification / Alternative check:
Perform a quick numerical check using approximations: tan 45 degrees ≈ 1 and sec 60 degrees ≈ 2. Then (4/√3) is approximately 4/1.732 ≈ 2.309, and (4/√3) × 2 ≈ 4.618. Adding tan 45 degrees gives roughly 5.618. Evaluating (√3 + 8)/√3 numerically gives (1.732 + 8)/1.732 ≈ 9.732/1.732 ≈ 5.618, which matches the approximate value of the original expression.

Why Other Options Are Wrong:
Option b ( (√3 + 8)/3 ) places 3 in the denominator instead of √3, changing the value significantly. Options c and d use (√3 - 8) instead of (√3 + 8), which would result in a negative or much smaller value. Option e ( (8 - √3)/2 ) does not correspond to the correct combination of tan and sec values and gives a different numerical result.

Common Pitfalls:
A common mistake is to misremember sec 60 degrees as 1/2 instead of 2, by confusing it with cos 60 degrees. Another pitfall is to attempt premature rationalisation of denominators, which can complicate the algebra unnecessarily before combining terms. Keeping the denominators as √3 until you combine like terms makes the arithmetic straightforward and reduces mistakes.

Final Answer:
The exact simplified value of tan 45 degrees + (4/√3) × sec 60 degrees is (√3 + 8)/√3.