cos²(π/8) - sin²(π/8)

Using the double angle formula for cosine:
cos(2x) = 2cos²(x) - 1
cos(π/4) = 2cos²(π/8) - 1
√2/2 = 2cos²(π/8) - 1
2cos²(π/8) = √2/2 + 1
cos²(π/8) = (√2/2 + 1)/2
cos²(π/8) = (√2 + 2)/4

Using the double angle formula for sine:
sin(2x) = 2sin(x)cos(x)
sin(π/4) = 2sin(π/8)cos(π/8)
√2/2 = 2sin(π/8)cos(π/8)
sin(π/8) = √2/4cos(π/8)

Substitute sin(π/8) back into the original equation:
cos²(π/8) - (sin(π/8))²
= cos²(π/8) - [(√2/4)cos(π/8)]²
= ([√2 + 2]/4) - [(√2/4)cos(π/8)]²
= ([√2 + 2]/4) - [(√2/4){√2/4cos(π/8)}]²
= ([√2 + 2]/4) - [(√2/4)²cos²(π/8)]
= ([√2 + 2]/4) - [(√2/4)²{√2 + 2}/4]
= ([√2 + 2]/4) - [√2/16(√2 + 2)]
= ([√2 + 2]/4) - [(√2 + 2)/4]
= [(√2 + 2) - (√2 + 2)]/4
= 0

Therefore, cos²(π/8) - sin²(π/8) = 0.