Question

Question #b954a

Answer 1

`0.`

Explanation:

We rearrange such of the terms

`cosalpha and cosbeta`, that, `alpha+beta=pi,` as shown below :

`:." The Exp.="{cos(2pi/9)+cos(7pi/9)}+{cos(4pi/9)+cos(5pi/9)}+{cos(pi/3)+cos(2pi/3)}`

`"Now, "alpha+beta=pi rArr cosalpha+cosbeta=cosalpha+cos(pi-alpha)`

`=cosalpha-cosalpha=0.`

`"Hence, the Exp.="{0}+{0}+{0}=0.`

Enjoy Maths.#

N.B.:

The Answer to the unedited Exp. is

`0+0+(-1)/3-cos(pi/3)=-1/3-1/2=-5/6.`

Answer 2

0

Explanation:

`cos 2pi/9 + cos 4 pi/9 + cos 5 pi/9 + cos 7 pi/9 + cos (pi/3) + cos 2 pi/3`

`=cos 40^@ + cos 80^@ + cos 100^@ + cos 140^@ + cos 60^@ + cos 120^@`

note:
`cos 100^@ = - cos 80^@, cos 140^@= - cos 40^@ and cos 120^@ = - cos 60^@`

`=cos 40^@ + cos 80^@ - cos 80^@ - cos 40^@ + cos 60^@ - cos 60^@ = 0`