If cot 60° + cosec 60° = x, use standard trigonometric ratios to find the exact numerical value of x.

Introduction / Context:
This question focuses on evaluating trigonometric expressions at special angles, here 60 degrees. Such problems check whether you remember exact values of sine, cosine, tangent, and their reciprocal functions at standard angles like 30 degrees, 45 degrees, and 60 degrees. Instead of using calculators, you must rely on memorised values or quick derivations from an equilateral triangle.

Given Data / Assumptions:
- The angle is 60 degrees, a standard angle in trigonometry.
- The expression is cot 60° + cosec 60° and this is equal to x.
- We must compute the exact simplified value of x using known trigonometric ratios.

Concept / Approach:
The key idea is to recall that cot is the reciprocal of tan, and cosec is the reciprocal of sin. For 60 degrees, sin 60° and tan 60° have well known exact values based on an equilateral triangle with side 2, where height is √3. Using these, we find cot 60° and cosec 60° separately and then add them.

Step-by-Step Solution:
Step 1: Recall that sin 60° = √3 / 2. Step 2: Since tan 60° = √3, the reciprocal relationship gives cot 60° = 1 / tan 60° = 1 / √3. Step 3: Cosecant is the reciprocal of sine, so cosec 60° = 1 / sin 60° = 1 / (√3 / 2) = 2 / √3. Step 4: Add these two values: cot 60° + cosec 60° = 1 / √3 + 2 / √3. Step 5: Combine the fractions over the common denominator √3 to get (1 + 2) / √3 = 3 / √3. Step 6: Simplify 3 / √3 by multiplying numerator and denominator by √3 if desired, which results in (3√3) / 3 = √3. Step 7: Therefore x = √3.
Verification / Alternative check:
You can verify the result approximately using decimal values. sin 60° is about 0.866, so cosec 60° is approximately 1.155. Tan 60° is about 1.732, so cot 60° is about 0.577. Adding them gives approximately 1.732, which matches the decimal value of √3 to three decimal places. This confirms the exact value is correct.

Why Other Options Are Wrong:
Option a and option b are more complicated expressions that do not simplify to √3 and are designed as distractors involving irrational numbers.
Option c, 1, is far too small given that cosec 60° alone is already greater than 1.
Option e, −√3/2, has the wrong sign and magnitude and does not match the positive value obtained from adding two positive trigonometric values.

Common Pitfalls:
Common mistakes include using incorrect standard values such as sin 60° = 1/2 or tan 60° = 1, which belong to other angles. Another error is to forget that cot is the reciprocal of tan and instead treat it as tan 30°. Confusing radian and degree measures is also possible but does not arise here because the question clearly uses degrees. Always double check standard angle values and reciprocal relationships to avoid these errors.

Final Answer:
The exact value of x = cot 60° + cosec 60° is √3.