2cos(x)(2cos(x-1)^2)+1=2sin(x)^2

They are pretty cool graphs Chris.     

I'm dazzled by this -- it can be solved by graphing.

You can make the substitution  1 - 2sin²(x)  =  cos(2x)

to get:  4cos(x)cos²(x-1) + cos(2x)  =  0,

but ...

2cos(x)(2cos(x-1)^2)+1=2sin(x)^2

The left side function is certainly a strange one........there are two solutions on [0, 360] ....one at 60.4° and the other at 300.4°..........here is the graph.......https://www.desmos.com/calculator/whgq6txzmz