If cosec 45° minus sin 30° equals x, then what is the exact value of x?

Introduction / Context:
This problem tests basic knowledge of standard trigonometric values for special angles, specifically 30 degrees and 45 degrees. It also checks the ability to perform simple arithmetic operations with surds. Such questions are common in aptitude and entrance exams as quick checks of trigonometric fluency.

Given Data / Assumptions:

• x = cosec 45° - sin 30°
• We use standard values: sin 30° and sin 45°.
• cosec θ is the reciprocal of sin θ.

Concept / Approach:
We recall:

• sin 30° = 1/2
• sin 45° = √2 / 2
• cosec θ = 1 / sin θ

First compute cosec 45°, then subtract sin 30°, and finally simplify the expression to match one of the given options.

Step-by-Step Solution:
sin 45° = √2 / 2 So cosec 45° = 1 / sin 45° = 1 / (√2 / 2) = 2 / √2 = √2 Also, sin 30° = 1/2 Now x = cosec 45° - sin 30° = √2 - 1/2 To compare with options, write √2 with denominator 2: √2 = (2√2) / 2 Hence x = (2√2)/2 - 1/2 = (2√2 - 1) / 2

Verification / Alternative check:
Use approximate values: √2 is about 1.414. Then x is 1.414 - 0.5 = 0.914. Now evaluate option d numerically: (2√2 - 1)/2 is (2*1.414 - 1)/2 = (2.828 - 1)/2 = 1.828 / 2 = 0.914, which matches, confirming that option d is correct.

Why Other Options Are Wrong:
Option a, 5/√3, is larger than 2 and does not match the approximate value of 0.914. Options b and c contain different combinations of surds and simple numbers, which do not reduce to √2 - 1/2 when simplified. Option e equals √2 - 1/2 in unsimplified form, but the problem expects the fully simplified expression that corresponds exactly to one of the listed options, which is given by option d.

Common Pitfalls:
Students sometimes misremember sin 30° or sin 45°, or they forget that cosec θ is the reciprocal of sin θ. Another error is mishandling rationalization or combining fractions with different denominators. Writing intermediate steps clearly and converting all terms to a common denominator helps avoid these mistakes.

Final Answer:
The exact value of x is (2√2 - 1)/2.