Question

Proove Cos5π/8 = -Cos3π/8

Answer 1

Prove that ( 1+ cos π/8)( 1+ cos 3π/8)( 1+ cos 5π/8)( 1+ cos 7π/8) =1/8?

(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.

Answer 2

ans=
cos5×180/8
=cos225/2.....(1)




-cos3×180/8
=-cos135/2
=cos(180-135/2)
=cos225/2.....(2)
(1)=(2)