Using exact trigonometric values, evaluate the expression sin 30° − cosec 45° and simplify your answer in surd form.

Introduction / Context:
This problem asks you to combine the sine of 30° with the cosecant of 45°, using exact trigonometric values. It is a straightforward exercise in recalling standard angle values, taking reciprocals for cosecant, and simplifying the resulting expression into a neat surd form.

Given Data / Assumptions:

• Expression: sin 30° − cosec 45°.
• Angles are in degrees.
• Standard values: sin 30°, sin 45°.
• cosec θ = 1 / sin θ.

Concept / Approach:
We first recall sin 30° and sin 45°, then compute cosec 45° as the reciprocal of sin 45°. Substituting these exact values into the expression gives a difference of a simple fraction and a surd. We then combine them over a common denominator to express the result as a single fraction in simplest surd form.

Step-by-Step Solution:
Recall sin 30° = 1/2. Also sin 45° = √2/2. Then cosec 45° = 1 / sin 45° = 1 / (√2/2) = 2/√2 = √2. Now compute sin 30° − cosec 45° = 1/2 − √2. Write √2 with denominator 2: √2 = (2√2)/2. So 1/2 − √2 = 1/2 − (2√2)/2 = (1 − 2√2)/2.

Verification / Alternative check:
Check numerically for reassurance. sin 30° ≈ 0.5 and cosec 45° ≈ 1.414. Their difference is approximately 0.5 − 1.414 = −0.914. Now evaluate (1 − 2√2)/2: 2√2 ≈ 2(1.414) ≈ 2.828, so numerator is 1 − 2.828 ≈ −1.828. Dividing by 2 gives about −0.914, matching the numerical calculation and confirming the simplified form.

Why Other Options Are Wrong:
Expressions like (2√6 − 1)/√3 or (√2 − √3)/√6 involve different combinations of surds that evaluate to values with different magnitudes from −0.914. The option (1 − 2√3)/2 would be much more negative because 2√3 is larger than 2√2. The expression (1 − √2)/2 corresponds to subtracting only one √2 instead of two in the numerator and does not match the exact value we computed. Only (1 − 2√2)/2 equals 1/2 − √2 exactly.

Common Pitfalls:
Students sometimes confuse cosec 45° with sec 45° or forget to take the reciprocal of sin 45°. Another mistake is to combine 1/2 and √2 incorrectly, for example adding instead of subtracting or failing to place them over a common denominator. Carefully using the definitions and standard values and performing the subtraction with a common denominator eliminates these errors.

Final Answer:
Therefore, the exact value of sin 30° − cosec 45° is (1 − 2√2)/2.