HOW TO PROVE(1+COS180/8)(1+COS3*180/8)(1+COS5*180/8)(1+COS7*180/8)=1/8

Answer 1

Or, in more usual notation,

(1+cos π/8)(1+cos 3π/8)(1+cos5π/8)(1+cos7π/8) = 1/8

cos 7π/8 = -cos π/8
cos 5π/8 = -cos 3π/8

So, you want

(1+cos π/8)(1-cos π/8)(1+cos 3π/8)(1-cos 3π/8)
= sin^2(π/8) sin^2(3π/8)

now, since π/8 = π/2 - 3π/8

= sin^2 π/8 cos^2 π/8
= (sin π/8 cos π/8)^2
= (1/2 sin π/4)^2
= (1/2 1/√2)^2
= 1/8