Difficulty: Easy
Correct Answer: √2
Explanation:
Introduction / Context:
This question checks basic trigonometry in a right triangle, especially how angles relate and how to evaluate standard angles exactly. Since ΔDEF is right angled at E, the two acute angles D and F must add up to 90°. When one acute angle is 45°, the other is also 45°, creating a 45°-45°-90° triangle. In such a triangle, sin 45°, cosec 45°, tan 45°, and cot 45° are all standard values. The expression cosec F × cot D becomes a direct substitution problem once the angles are identified correctly.
Given Data / Assumptions:
Concept / Approach:
Use triangle angle sum in a right triangle:
∠D + ∠F = 90°.
Then evaluate:
cosec 45° = 1 / sin 45°
cot 45° = 1 / tan 45°.
Finally multiply the exact values.
Step-by-Step Solution:
1) Since ∠E = 90°, the remaining angles satisfy:
∠D + ∠F = 90°
2) Given ∠D = 45°:
∠F = 90° - 45° = 45°
3) Compute cosec F:
cosec 45° = 1 / (sin 45°) = 1 / (1/√2) = √2
4) Compute cot D:
cot 45° = 1
5) Multiply:
cosec F × cot D = √2 * 1 = √2
Verification / Alternative check:
In a 45°-45°-90° triangle, the legs are equal and the hypotenuse is √2 times a leg. That geometry implies sin 45° = 1/√2, hence cosec 45° = √2. Also tan 45° = 1, hence cot 45° = 1. The product must therefore be √2, consistent with the computed result.
Why Other Options Are Wrong:
• 2: would require cosec 45° to be 2, which is incorrect.
• 1/2 or 1/√2: come from confusing sin with cosec or taking reciprocal in the wrong direction.
• 1: would be true only if cosec 45° were 1, which it is not.
Common Pitfalls:
• Forgetting that the two acute angles in a right triangle add to 90°.
• Mixing up cosec and sec, or forgetting cosec is the reciprocal of sin.
• Misremembering sin 45° as √2/2 but not converting correctly to cosec.
Final Answer:
√2
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