Question

Prove that cos(2pi/15)*cos(4pi/15)*cos(8pi/15)*cos(16pi/15)=1/16

Answer 1

it's true ! but idk how to prove

Answer 2

because i have no idea how to write it on here

Answer 3

If you know what each of the cos(xπ/15) equals you could write those down being multiplied and then do out the work

Answer 4

sin(a+b)= sin(a)cos(b)+sin(b)cos(a) (1)
for a=b, sin(2a)= 2 sin(a) cos(a) (2)

let a= 16 pi/15 (3)
(so 2a = 32 pi/15 )

then using (3) in (2), we have
sin(2a)= 2 sin(a) cos(a)
= 2 (2 sin(a/2) cos(a/2)) cos(a)
= 2 (2 (2 sin(a/4) cos(a/4)) cos(a/2)) cos(a)
= 2 (2 (2 (2 sin(a/8) cos(a/8)) cos(a/4)) cos(a/2)) cos(a)
= 16 sin(a/8) ( cos(a/8) cos(a/4) cos(a/2) cos(a))

now note sin(2a) = sin(2 pi/15) and sin(a/8)= sin(2 pi/15)
so,
cos(a/8) cos(a/4) cos(a/2) cos(a)= 1/16
or, replacing a with 16 pi/15,
cos(2pi/15)*cos(4pi/15)*cos(8pi/15)*cos(16pi/15)=1/16