Question

if A=22.5 than (1+cosA)(1+cos3A)(1+cos5A)(1+cos7A)=?

Answer 1

(1+cos A)(1+cos 7A) = 1+ cos A + cos 7A + cos A cos 7A
= 1+ 2 cos 4A cos 3A + 1/2 * (cos 8A + cos 6A)
= 1 + 0 + 1/2 (-1 -1/sqrt2)
as 4A= pi/2, 8A= pi, 6A=3pi/4
.
Similarly
(1+cos 3A) (1+ cos 5A)
= 1 + cos 3A + cos 5A + cos 3A cos 5A
= 1 + 2 cos A cos 4A + 1/2 (cos 2A + cos 8A)
= 1 + 0 + 1/2 (1/sqrt2 - 1)
.
Multiply the factors to get
(1/2)^2 - (1/2sqrt2)^2
= 1/4 - 1/8 = 1/8