Find the value of (sin²θ - 2sin⁴θ) / (2cos⁴θ + cos²θ) at θ = 45°.

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Step 1: Substitute θ = 45°.

We know sin 45° = cos 45° = √2/2.

Step 2: Calculate sin²θ and sin⁴θ:
- sin²45° = (√2/2)² = 1/2
- sin⁴45° = (sin²45°)² = (1/2)² = 1/4

Step 3: Calculate cos²θ and cos⁴θ:
- cos²45° = (√2/2)² = 1/2
- cos⁴45° = (cos²45°)² = (1/2)² = 1/4

Step 4: Substitute these values into the expression:

Numerator = sin²θ - 2sin⁴θ = (1/2) - 2*(1/4) = (1/2) - (1/2) = 0  
Denominator = 2cos⁴θ + cos²θ = 2*(1/4) + (1/2) = (1/2) + (1/2) = 1

Step 5: Calculate the value:

Value = Numerator / Denominator = 0 / 1 = 0

Therefore, the value of the expression at θ = 45° is 0.

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