In right-angled triangle ΔDEF with ∠E = 90 degrees, if cos D = 8/17, use complementary angles and trigonometric identities to find the exact value of sec F and choose the correct option.

Introduction / Context:
This question connects trigonometric ratios of complementary angles in a right triangle. You are given cos D in triangle ΔDEF where angle E is 90 degrees, and you must determine sec F. Since D and F are the acute angles in a right triangle, they are complementary, which allows you to use relationships between their sine and cosine values.

Given Data / Assumptions:

• Triangle ΔDEF is right angled at E, so ∠E = 90 degrees.
• cos D = 8/17.
• Angles D and F are complementary: ∠D + ∠F = 90 degrees.
• sec F = 1 / cos F.

Concept / Approach:
In a right triangle, if one acute angle is D and the other is F, then F = 90 degrees - D. This implies cos F = sin D and sin F = cos D. Once you know cos D, you can find sin D using the Pythagorean identity sin^2 D + cos^2 D = 1. Then cos F = sin D, and sec F is simply the reciprocal of cos F.

Step-by-Step Solution:
Given cos D = 8/17, compute sin D from sin^2 D + cos^2 D = 1. cos^2 D = (8/17)^2 = 64/289. Then sin^2 D = 1 - 64/289 = (289 - 64)/289 = 225/289. Since D is acute, sin D is positive, so sin D = 15/17. Because F is complementary to D, cos F = sin D = 15/17. Therefore sec F = 1 / cos F = 1 / (15/17) = 17/15.
Verification / Alternative check:
You can imagine a right triangle with hypotenuse 17 and adjacent side to angle D equal to 8, so the opposite side to D is 15 by the 8–15–17 Pythagorean triplet. In that picture, cos D = 8/17, sin D = 15/17, and cos F, being the cosine of the complementary angle, equals 15/17. Taking its reciprocal again yields sec F = 17/15, confirming the calculation.

Why Other Options Are Wrong:
Option a ( 8/17 ) is cos D, not sec F. Option b ( 15/17 ) is cos F, not its reciprocal. Option d ( 17/8 ) is sec D rather than sec F. Option e ( 15/8 ) would correspond to tan D and does not match the required function.

Common Pitfalls:
Learners sometimes assume that sec F equals sec D or confuse which side is adjacent to which angle in the triangle. Another common problem is misusing the Pythagorean identity by subtracting in the wrong direction. Clearly labelling sides and angles and remembering the complementary relationship between D and F helps avoid these mistakes.

Final Answer:
The exact value of sec F in the given right triangle is 17/15.