In basic trigonometry with angles in degrees, evaluate the expression Tan 45° + Cosec 60° and write the result in exact surd form (no decimals).

Introduction / Context:
This question combines two standard trigonometric values, Tan 45° and Cosec 60°, and asks you to express their sum in exact surd form. Such problems check your recall of special angle values and your ability to handle square roots and fractions correctly. These skills are widely used in geometry, trigonometry, and many aptitude tests.

Given Data / Assumptions:

• The expression to evaluate is Tan 45° + Cosec 60°.
• Angles are measured in degrees.
• We should use exact values for the trigonometric functions, not decimal approximations.
• The final answer must be presented in simplified surd form.

Concept / Approach:
First, recall the exact values of the tangent and sine functions at 45° and 60°. For 45°, tangent has a particularly simple value, and for 60°, the sine value leads directly to the cosecant. After substituting these values, we add them and if necessary rationalise or combine the terms into a single fraction to compare with the answer choices. No advanced identities are required, just standard special angle facts and simple arithmetic.

Step-by-Step Solution:
1) Recall that Tan 45° = 1. 2) For 60°, sin 60° = √3 / 2, so Cosec 60° = 1 / sin 60° = 2 / √3. 3) Substitute into the expression: Tan 45° + Cosec 60° = 1 + 2 / √3. 4) Rewrite 1 as √3 / √3 to combine with 2 / √3: 1 = √3 / √3. 5) Add the fractions: √3 / √3 + 2 / √3 = (√3 + 2) / √3. 6) To express this in a rationalised form, multiply numerator and denominator by √3: (√3 + 2) / √3 * (√3 / √3) = ( (√3 + 2)√3 ) / 3. 7) Expand the numerator: (√3 + 2)√3 = 3 + 2√3. 8) So the final simplified surd form is (3 + 2√3) / 3.

Verification / Alternative check:
As a numerical check, approximate √3 ≈ 1.732. Then Tan 45° + Cosec 60° ≈ 1 + 2 / 1.732 ≈ 1 + 1.155 ≈ 2.155. Now evaluate (3 + 2√3) / 3 numerically: (3 + 2 * 1.732) / 3 = (3 + 3.464) / 3 = 6.464 / 3 ≈ 2.155. The two approximations match, confirming that the symbolic simplification is consistent with the numeric value.

Why Other Options Are Wrong:
Option b (5 / √3) simplifies to approximately 2.887, which is too large. Option c, option d, and option e yield different approximate values when substituting √3 ≈ 1.732 and none match 2.155. Only option a, (3 + 2√3) / 3, matches both the algebraic derivation and the numeric approximation of Tan 45° + Cosec 60°.

Common Pitfalls:
A common mistake is to use incorrect special angle values, such as taking sin 60° = 1/2 instead of √3 / 2, which leads to the wrong cosecant. Another pitfall is failing to rationalise the denominator consistently when combining 1 and 2 / √3, resulting in forms that do not match any of the given options. Practising the core special angle values and simple surd manipulations makes this type of question very manageable.

Final Answer:
The exact surd value of Tan 45° + Cosec 60° is (3 + 2√3)/3.