How to simply: $\prod_{k=1}^{100}(1+2cos\frac{2\pi.3^k}{3^{100} +1})$?

Question

$$\prod_{k=1}^{100}\left(1+2\cos\left(\frac{2\pi.3^k}{3^{100} +1}\right)\right)$$ equals

.I tried do this problem many time but i don't figure outhow to do this problem .

Answer 1

HINT:

For $\sin y\ne0,$

$$1+2\cos2y=1+2(1-2\sin^2y)=\dfrac{\sin3y}{\sin y}$$

Observe the Telescoping nature and use $\sin(\pi-u)=+\sin u$

Answer 2

$$1+2\cos2y_k=1+2(1-2\sin^2y_k)=\dfrac{\sin3y_k}{\sin y_k}$$ $$y_k=\frac{3^k\pi}{3^{100} +1}$$ Product will reduce to $$\dfrac{\sin3y_{100}}{\sin y_1}$$ Just observe $$\dfrac{\sin(3\pi - y_{1})}{\sin y_1}$$