## anonymous 4 years ago What is the ordered pair of positive intgers (a,b) for which a/b is a reduced fraction and $x = \frac{a \pi}{b}$ is the least positive solution of the equation: $(2\cos(8x)−1)(2\cos(4x)−1)(2\cos(2x)−1)(2\cos(x)−1)=1$

1. asnaseer

From inspection ($$x=0$$) (and therefore $$x=2n\pi$$, $$n=0,1,2,...$$) would certainly be a solution to this, but can you confirm that this is not valid since you said "least POSITIVE solution" and I understand that to mean $$x\gt 0$$?

2. anonymous

Then other than 0, are there any other possible pairs?

3. anonymous

Well we know for cos(2ax) to be 1 we need that: $(2m)\frac{a \pi}{b}=2n \pi; n,m \in \mathbb{Z}$ $a=\frac{bn}{m}; n,m \in \mathbb{Z}$ I'm not sure if you might be looking for a different form.