What is the value of the expression (4/3)cot²(π/6) + 3cos² 150° − 4cosec² 45° + 8sin(π/2)?

Introduction / Context:
This question asks you to evaluate a mixed trigonometric expression involving cotangent, cosine, cosecant and sine, at specific standard angles. It tests recall of exact trigonometric values for special angles such as 30°, 45°, 90° and 150°, and the ability to handle squares of these functions accurately.

Given Data / Assumptions:
- Expression: (4/3)cot²(π/6) + 3cos² 150° − 4cosec² 45° + 8sin(π/2).
- π/6 corresponds to 30° and π/2 corresponds to 90°.
- All angles are in the standard sense used in trigonometry, and we use their well known exact values.

Concept / Approach:
We evaluate each trigonometric term separately using special angle values, then substitute into the expression. Squaring must be done carefully so that signs are handled correctly. Finally, we add and subtract the numeric results to find the final simplified value.

Step-by-Step Solution:
Step 1: For π/6 (30°), tan 30° = 1/√3, so cot 30° = √3. Thus cot²(π/6) = (√3)² = 3.Step 2: Compute (4/3)cot²(π/6) = (4/3) * 3 = 4.Step 3: For 150°, we have cos 150° = cos(180° − 30°) = −cos 30° = −√3/2. So cos² 150° = (−√3/2)² = 3/4.Step 4: Then 3cos² 150° = 3 * (3/4) = 9/4.Step 5: For 45°, sin 45° = √2/2, so cosec 45° = 1 / sin 45° = √2. Then cosec² 45° = 2, and −4cosec² 45° = −4 * 2 = −8.Step 6: For π/2 (90°), sin(π/2) = sin 90° = 1, so 8sin(π/2) = 8 * 1 = 8.Step 7: Now add all contributions: first two terms give 4 + 9/4 = (16/4 + 9/4) = 25/4.Step 8: The last two terms are −8 + 8 = 0, so they cancel out.Step 9: Therefore, the value of the entire expression is 25/4.

Verification / Alternative check:
A quick consistency check is to approximate each trigonometric value in decimal form, evaluate the expression numerically, and confirm that the result is close to 6.25, which equals 25/4. This reassures us there were no sign or arithmetic errors in the exact calculation.

Why Other Options Are Wrong:
The options 1, −7/2 and 13/2 arise from miscalculations such as forgetting to square before multiplying, using the wrong sign for cos 150°, or mishandling the −4 factor with cosec² 45°. Since the −8 and +8 terms exactly cancel, any answer not equal to 25/4 indicates some step was not handled correctly.

Common Pitfalls:
Students frequently forget that cos 150° is negative, leading to sign errors. Confusion between cosec and sec, or between tan and cot, is another common issue. Also, mixing degrees and radians without noticing can create mistakes, but in this problem the special angles correspond cleanly, so using the standard values is safe.

Final Answer:
The value of the given expression is 25/4.