Use standard trigonometric values at special angles (30 degrees and 60 degrees) to evaluate exactly.\nFind the exact value of: cosec^2 60 degrees + sec^2 60 degrees + tan^2 30 degrees.

Introduction / Context:
This question tests exact evaluation of trigonometric functions at special angles. The key is to use known values for sin 60 degrees, cos 60 degrees, and tan 30 degrees, then square them as required and add carefully using fractions. These are common simplification steps in trigonometry and aptitude problems.

Given Data / Assumptions:

• Expression: cosec^2 60 + sec^2 60 + tan^2 30
• Use exact special-angle values (no rounding).
• cosec theta = 1/sin theta, sec theta = 1/cos theta.

Concept / Approach:
Compute each term separately using standard exact values:\nsin 60 = √3/2, cos 60 = 1/2, tan 30 = 1/√3.\nThen square cosec, sec, and tan as asked and add using common denominators.

Step-by-Step Solution:
sin 60 = √3/2 => cosec 60 = 1/(√3/2) = 2/√3 So cosec^2 60 = (2/√3)^2 = 4/3 cos 60 = 1/2 => sec 60 = 1/(1/2) = 2 So sec^2 60 = 2^2 = 4 tan 30 = 1/√3 So tan^2 30 = (1/√3)^2 = 1/3 Add: 4/3 + 4 + 1/3 = (5/3) + 4 Convert 4 to thirds: 4 = 12/3 Total = (5/3 + 12/3) = 17/3 = 5 2/3

Verification / Alternative check:
Combine the fraction terms first: 4/3 + 1/3 = 5/3. Then adding 4 gives 5/3 + 12/3 = 17/3. This confirms the arithmetic cleanly.

Why Other Options Are Wrong:
5 ignores part of the fractional contribution (5/3).
6 is too large and typically comes from treating cosec^2 60 as 4 instead of 4/3.
5 1/2 and 5 1/3 result from incorrect fraction addition or rounding √ values instead of using exact squares.

Common Pitfalls:
Forgetting to square after computing cosec/sec/tan, confusing tan 30 with tan 60, and converting between mixed numbers and improper fractions incorrectly.

Final Answer:
cosec^2 60 + sec^2 60 + tan^2 30 = 5 2/3