Let theta be an acute angle where 0° lt theta lt 90°. Consider the expression \(\sin \theta \cdot \cos(90° -...

1. TRANSLATE the problem information

• Given: \(\sin \theta \cdot \cos(90° - \theta) - \cos \theta \cdot \sin(90° - \theta)\) where \(0° \lt \theta \lt 90°\)
• Find: Which expression is equivalent from the given choices

2. INFER the approach

• The presence of complementary angles \((90° - \theta)\) suggests using cofunction identities
• Alternatively, the pattern 'sin a cos b − cos a sin b' matches the sine difference formula
• Either approach should lead to the same simplified form

3. SIMPLIFY using cofunction identities

• Apply \(\cos(90° - \theta) = \sin \theta\) and \(\sin(90° - \theta) = \cos \theta\)
• Substitute: \(\sin \theta \cdot \sin \theta - \cos \theta \cdot \cos \theta = \sin²\theta - \cos²\theta\)

4. INFER the connection to double angle formulas

• Recognize that \(\cos(2\theta) = \cos²\theta - \sin²\theta\)
• Therefore: \(\sin²\theta - \cos²\theta = -(\cos²\theta - \sin²\theta) = -\cos(2\theta)\)

Answer: (D) \(-\cos(2\theta)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that complementary angle expressions should trigger cofunction identity usage. Instead, they might try to work with the original expression directly or attempt to use other trigonometric identities inappropriately. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly apply cofunction identities to get \(\sin²\theta - \cos²\theta\), but then fail to properly connect this to the double angle identity. They might forget that \(\cos(2\theta) = \cos²\theta - \sin²\theta\) and miss the negative sign, potentially selecting Choice (B) \(\cos(2\theta)\).

The Bottom Line:

This problem tests the ability to recognize patterns that suggest specific trigonometric identities, particularly cofunction relationships and double angle formulas. Success requires both pattern recognition and careful algebraic manipulation with attention to signs.