In right triangle DEF, angle E is 90°. If m∠D = 45°, then what is the exact value of cosec F, where F is the remaining acute angle of the triangle?

Introduction / Context:
This question involves basic properties of right triangles and trigonometric ratios of standard angles. When one acute angle of a right triangle is known, the other acute angle is determined, and we can use known sine values to compute cosecant. Here we work with a 45°–45°–90° triangle where both acute angles are equal.

Given Data / Assumptions:

• Triangle DEF is right angled at E, so m∠E = 90°.
• m∠D = 45°.
• Angles of a triangle sum to 180°.
• cosec θ = 1 / sin θ.

Concept / Approach:
Since the triangle is right angled at E and one acute angle is 45°, the third angle must also be 45° because 90° + 45° + 45° = 180°. That means angles D and F are both 45°. The sine of 45° is √2/2, so the cosecant of 45° is its reciprocal, which simplifies to √2. We simply apply these standard ratios to find cosec F.

Step-by-Step Solution:
Sum of angles: m∠D + m∠E + m∠F = 180°. Given m∠E = 90° and m∠D = 45°. So m∠F = 180° − 90° − 45° = 45°. We need cosec F = cosec 45°. Recall sin 45° = √2/2. Hence cosec 45° = 1 / sin 45° = 1 / (√2/2) = 2/√2 = √2. Therefore cosec F = √2.

Verification / Alternative check:
Consider a 45°–45°–90° triangle with legs of length 1 and hypotenuse √2. For angle F, the sine is opposite/hypotenuse = 1/√2 = √2/2. The reciprocal of this value is √2, which matches the calculated cosec F. This geometric picture confirms the result derived from standard trigonometric values.

Why Other Options Are Wrong:
1/2 and 1/√2 are values associated with sine or cosine of 30° or 45°, not with cosecant. 1/√3 corresponds to trigonometric ratios for 30° or 60°, not for 45°. The value 2 would be cosec 30° in a 30°–60°–90° triangle, not in a 45°–45°–90° triangle. Only √2 matches the exact value of cosec 45° and therefore cosec F.

Common Pitfalls:
Some learners mistakenly assume that if D is 45°, then F might be 30° or 60°, forgetting that E is already 90°. Others mix up the roles of sine and cosecant or fail to simplify 2/√2 to √2. Remember that the reciprocal of √2/2 is √2 and that in a right triangle the two acute angles must sum to 90°.

Final Answer:
Therefore, the value of cosec F is √2.