In right-angled triangle ΔABC with ∠B = 90 degrees and ∠A = 45 degrees, determine the exact value of the expression cosec C + 1/√3 using basic trigonometry, and then choose the correct option.

Introduction / Context:
This question combines knowledge of right triangles with standard trigonometric values. You are given a right-angled triangle ΔABC where one acute angle is 45 degrees. The other acute angle can be deduced, and then you must evaluate cosec of that angle and add 1/√3, finally matching the result with one of the given expressions.

Given Data / Assumptions:

• Triangle ΔABC is right angled at B, so ∠B = 90 degrees.
• ∠A = 45 degrees.
• The sum of interior angles of a triangle is 180 degrees.
• cosec θ = 1 / sin θ for any angle θ where sine is defined.

Concept / Approach:
First find angle C by using the angle sum property of triangles. Then recall the standard sine value for that angle and compute its cosecant. Finally add 1/√3 to this cosecant and simplify the result into a single surd expression. Then compare the simplified expression with the options provided.

Step-by-Step Solution:
Use the angle sum property: ∠A + ∠B + ∠C = 180 degrees. Substitute the known angles: 45 + 90 + ∠C = 180, so ∠C = 45 degrees. For 45 degrees, sin 45 degrees = √2/2, so cosec 45 degrees = 1 / (√2/2) = √2. Compute the expression cosec C + 1/√3 = √2 + 1/√3. Rewrite √2 as √6/√3 so that the terms have a common denominator: √2 = √6/√3. Thus √2 + 1/√3 = (√6/√3) + (1/√3) = (√6 + 1)/√3.
Verification / Alternative check:
You may check numerically by approximating √2 and √3. For example, √2 is about 1.414 and √3 is about 1.732, so √2 + 1/√3 ≈ 1.414 + 0.577 ≈ 1.991. Evaluating (√6 + 1)/√3 gives approximately the same value, confirming that the algebraic simplification is correct and corresponds to option d.

Why Other Options Are Wrong:
Option a ( (2 + √3)/2 ) and option b ( 5/√3 ) have different numeric values and do not match √2 + 1/√3. Option c ( (3√2 + 1)/3 ) and option e ( (√2 + √3)/3 ) mix surds incorrectly and produce different approximations. None of these equals the exact expression obtained from the triangle properties and trigonometric definitions.

Common Pitfalls:
A frequent error is to miscalculate angle C as something other than 45 degrees or to confuse cosec with sec. Another pitfall is to add the surds without bringing them to a common denominator, which leads to mismatched expressions when compared with the options. Carefully rewriting √2 in terms of √3 is essential for correct simplification.

Final Answer:
The exact value of cosec C + 1/√3 is (√6 + 1)/√3.