If tan 45° + cosec 30° = x, then what is the numerical value of x?

Introduction / Context:
This problem is about using standard trigonometric values for special angles. We are given an expression involving tan 45 degrees and cosec 30 degrees, asked to evaluate it, and express the answer as a simple number. Knowledge of exact values at 30, 45 and 60 degrees is very important for many aptitude questions.

Given Data / Assumptions:

• The expression is tan 45° + cosec 30°.
• We are told this equals x, and we must find x.
• We can use the usual exact trigonometric values without approximation.

Concept / Approach:
We recall the standard values: tan 45° = 1 and sin 30° = 1/2, so cosec 30° which is the reciprocal of sin 30° equals 2. Substituting these values into the expression tan 45° + cosec 30° allows immediate evaluation without any complicated simplification. The key is to remember the reciprocal relationships and not to confuse sine with cosine.

Step-by-Step Solution:
Step 1: Recall that tan 45° = 1. Step 2: Recall that sin 30° = 1/2. Step 3: Since cosec 30° is 1 / sin 30°, we have cosec 30° = 1 / (1/2) = 2. Step 4: Substitute into the expression: tan 45° + cosec 30° = 1 + 2. Step 5: Compute the sum: 1 + 2 = 3. Step 6: Therefore x = 3.

Verification / Alternative check:
We can check by using decimal approximations, although they are not necessary. tan 45° is exactly 1 and sin 30° is exactly 0.5, so cosec 30° is 2. The sum 1 + 2 is obviously 3. There is no ambiguity here, so the value of x is clearly established.

Why Other Options Are Wrong:

• Option A: √3 is approximately 1.732 and does not match 3.
• Option B: (1 - 2√2)/√2 is a negative quantity, very different from 3.
• Option C: (√3 - 4)/(2√3) is a small negative fraction.
• Option E: 4 is larger than the correct sum and has no basis from the given values.

Common Pitfalls:
Students sometimes confuse cosec with sec and may use the reciprocal of cos 30° instead of sin 30°. Another mistake is to think tan 45° equals √3, which is actually the value of tan 60°. Memorising the basic triangle with angles 30°, 60°, and 90° and the isosceles right triangle for 45° helps avoid these errors.

Final Answer:
Thus, the value of x is 3.