Question

Find the value of 5Cdfrac%7B3%5Cpi%7D%7B9%7D%5Cright%29%2B%5Ccos%5Cleft%28%5Cdfrac%7B5%5Cpi%7D%7B9%7D%5Cright%29%2B%5Ccos%5Cleft%28%5Cdfrac%7B7%5Cpi%7D%7B9%7D%5Cright%29" id="TexFormula1" title=" \cos\left(\dfrac{\pi}{9}\right)+\cos\left(\dfrac{3\pi}{9}\right)+\cos\left(\dfrac{5\pi}{9}\right)+\cos\left(\dfrac{7\pi}{9}\right)" alt=" \cos\left(\dfrac{\pi}{9}\right)+\cos\left(\dfrac{3\pi}{9}\right)+\cos\left(\dfrac{5\pi}{9}\right)+\cos\left(\dfrac{7\pi}{9}\right)" align="absmiddle" class="latex-formula">
Show Your work!:)

Answer 1

Recall the identity,

$\cos\theta+\cos3\theta+\cos5\theta+\cos7\theta=\dfrac{\sin8\theta}{2\sin\theta}$

(link to proof in the comments)

so that the required sum has a value of

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

Recall another identity,

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

which means

$\sin\dfrac{8\pi}9=\sin\left(\pi-\dfrac\pi9\right)=\sin\dfrac\pi9$

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