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Simplify and find the exact value of [2 * sin(45° + θ) * sin(45° - θ)] / cos 2θ.

Difficulty: Medium

Correct Answer: 1

Explanation:


Introduction / Context:
This problem tests product-to-sum identities and the ability to recognize a cancellation with cos 2θ. The angles (45° + θ) and (45° - θ) are conjugate pairs, so their product collapses to a simple cosine expression using sin u * sin v identity. The remaining division by cos 2θ completes the simplification.


Given Data / Assumptions:

  • Expression: [2*sin(45° + θ)*sin(45° - θ)]/cos 2θ
  • Identity: sin u * sin v = (cos(u - v) - cos(u + v))/2
  • cos 90° = 0


Concept / Approach:
Apply sin u * sin v identity, simplify u - v and u + v, then multiply by 2 and divide by cos 2θ. The structure is designed for exact cancellation.


Step-by-Step Solution:

Step 1: Let u = 45° + θ and v = 45° - θ. Step 2: sin u * sin v = (cos(u - v) - cos(u + v))/2. Step 3: u - v = (45° + θ) - (45° - θ) = 2θ. Step 4: u + v = (45° + θ) + (45° - θ) = 90°. Step 5: So sin u * sin v = (cos 2θ - cos 90°)/2 = (cos 2θ - 0)/2 = cos 2θ/2. Step 6: Multiply by 2: 2*sin u*sin v = 2*(cos 2θ/2) = cos 2θ. Step 7: Divide by cos 2θ: (cos 2θ)/(cos 2θ) = 1.


Verification / Alternative check:
Take θ = 0: numerator becomes 2*sin45*sin45 = 2*(1/sqrt(2))^2 = 1, and cos 0 = 1, ratio = 1. Matches the simplified result.


Why Other Options Are Wrong:

0: would require the numerator to be 0 for all θ, which is not true. tan 2θ or cot 2θ: these would appear if the numerator became sin 2θ, but it becomes cos 2θ. sec 2θ: would be 1/cos 2θ, not 1.


Common Pitfalls:
Using the wrong product identity (confusing sin u sin v with sin u cos v), or miscomputing u + v as 2*45° + 2θ instead of 90°.


Final Answer:
1

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