Question

Value of (1 +cosπ/8)(1 +cos3π/8)(1 +cos5π/8)(1 +cos7π/8)

Answer 1

1 + cos π/8)(1 + cos 3π/8)(1 + cos 5π/8)(1 + cos 7π/8)

= (1 + cos π/8)(1 + cos 3π/8)(1 - cos 3π/8)(1 - cos π/8),

since cos(π - t) = -cos t

= (1 - cos² π/8)(1 - cos² 3π/8)

= (1 - cos² π/8)(1 - sin² π/8),

via cos(π/2 - t) = sin t

= (1 - cos² π/8)(1 - sin² π/8)

= sin ^2 π/8 cos ^2 π/8

= (1/4) (2 sin π/8 cos π/8)²

= (1/4) (sin 2π/8)²
, by double angle formula
= (1/4) (sin π/4)²
= (1/4) (1/√2)²
= (1/4)(1/2)
= 1/8.