Find the exact value of the trigonometric expression sec²17° − 1/(tan²73°) − sin 17° · sec 73°.

Introduction / Context:
This question is designed to test your ability to use complementary angle relationships and basic trigonometric identities. The angles 17° and 73° add up to 90°, so they are complementary. Recognising this helps simplify the expression without resorting to calculator approximations.

Given Data / Assumptions:

• Expression: sec²17° − 1/(tan²73°) − sin 17° · sec 73°.
• Angles 17° and 73° satisfy 17° + 73° = 90°.
• Trigonometric functions are in degrees and lie in a standard context where identities apply.

Concept / Approach:
Key concepts:

• Complementary angle identities: tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.
• Pythagorean identities: sec²θ = 1 + tan²θ and 1 + cot²θ = cosec²θ.
• Use angle transformations to rewrite tan 73° and sec 73° in terms of 17°, then simplify step by step.

Step-by-Step Solution:
Step 1: Use complement relation: 73° = 90° − 17°. Step 2: Then tan 73° = tan(90° − 17°) = cot 17°. Step 3: So tan²73° = cot²17° and 1 / tan²73° = 1 / cot²17° = tan²17°. Step 4: Also, sec 73° = sec(90° − 17°) = cosec 17°. Step 5: Therefore sin 17° · sec 73° = sin 17° · cosec 17° = 1. Step 6: Rewrite the expression in terms of 17° only: sec²17° − tan²17° − 1. Step 7: Use identity sec²θ = 1 + tan²θ, so sec²17° − tan²17° = 1. Step 8: Thus the expression becomes 1 − 1 = 0.
Verification / Alternative check:
Step 1: One can approximate numerically: pick approximate values of tan 17°, sec 17°, sin 17° and tan 73°, sec 73° from tables or a calculator. Step 2: Compute each term and observe that sec²17° − 1/(tan²73°) is close to 1, while sin 17° · sec 73° is very close to 1. Step 3: Their difference is then extremely close to 0, confirming the exact symbolic simplification.
Why Other Options Are Wrong:
Option 1 would require sec²17° − tan²17° − 1 to simplify to 1, but we showed it simplifies to 0. Option −1 would imply sec²17° − tan²17° equals 0, contradicting the identity sec²θ = 1 + tan²θ. Option 2 or −2 would arise only if the identity or complementary relations were misapplied, which is not correct here. Therefore 0 is the only consistent result.
Common Pitfalls:
Students sometimes forget or misapply complementary angle formulas, for example confusing tan(90° − θ) with tan θ. Another error is to treat sec 73° incorrectly as 1 / sin 17° instead of 1 / cos 73°, which then simplifies to cosec 17°. Some learners try to approximate numerically early, which is slower and prone to rounding errors compared to using exact identities.
Final Answer:
The exact value of the expression is 0.