Find the value of \cos\left((\pi)/(9)\right)+\cos\left((3\pi)/(9)\right)+\cos\left((5\pi)/(9)\right)+\cos\left((7\pi)/(9)\right) Show Your work!:)

Question

Find the value of
`\cos\left((\pi)/(9)\right)+\cos\left((3\pi)/(9)\right)+\cos\left((5\pi)/(9)\right)+\cos\left((7\pi)/(9)\right)`

Show Your work!:)

Answer 1

Recall the identity,

`\cos\theta+\cos3\theta+\cos5\theta+\cos7\theta=(\sin8\theta)/(2\sin\theta)`

(link to proof in the comments)

so that the required sum has a value of

`\cos\frac\pi9+\cos\frac{3\pi}9+\cos\frac{5\pi}9+\cos\frac{7\pi}9=\frac{\sin\frac{8\pi}9}{2\sin\frac\pi9}`

Recall another identity,

`\sin(\pi-x)=\sin x`

which means

`\sin\frac{8\pi}9=\sin\left(\pi-\frac\pi9\right)=\sin\frac\pi9`

Then the sum's value reduces to 1/2.