Which of the following expressions is equivalent to \((\cos^2 18°) - (\cos^2 72°)\)? \(-\sin(54°)\) \(\cos(72°)\) \(\sin(36°)\) \(\sin(54°)\)...

1. INFER key angle relationships

• Given: \(\cos^2(18°) - \cos^2(72°)\)
• Key insight: \(18° + 72° = 90°\), so these angles are complementary
• This means we can use cofunction identities to simplify

2. INFER the first transformation strategy

• Since \(72° = 90° - 18°\), we can use: \(\cos(72°) = \sin(18°)\)
• This transforms our expression to: \(\cos^2(18°) - \sin^2(18°)\)

3. INFER pattern recognition for trigonometric identities

• The form \(\cos^2(θ) - \sin^2(θ)\) matches the double-angle identity
• Double-angle formula: \(\cos(2θ) = \cos^2(θ) - \sin^2(θ)\)
• Therefore: \(\cos^2(18°) - \sin^2(18°) = \cos(2 \times 18°) = \cos(36°)\)

4. INFER final conversion to match answer choices

• \(\cos(36°)\) doesn't appear directly in the choices
• Use cofunction identity: \(\cos(36°) = \sin(90° - 36°) = \sin(54°)\)
• This matches Choice D

Answer: D. \(\sin(54°)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(18°\) and \(72°\) are complementary angles

Students might try to work with the cosines directly without seeing the \(90°\) relationship. Without this key insight, they get stuck trying to evaluate exact values or attempt incorrect algebraic manipulations. This leads to confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Stopping at \(\cos(36°)\) without converting to match answer choices

Students correctly derive \(\cos(36°)\) but don't realize they need to use cofunction identities again to find an equivalent expression among the given options. They might incorrectly think \(\cos(36°) = \sin(36°)\) and select Choice C (\(\sin(36°)\)).

The Bottom Line:

This problem requires multiple applications of cofunction identities and pattern recognition for trigonometric formulas. Success depends on seeing angle relationships and strategically applying identities in sequence rather than trying to compute exact decimal values.