If sqr(2)cosx-1=(1+sqr3)/2 and sqr(2)cosx+1=(1-sqr3)/2, find the value of cos4x

Question

The answer is -1/2. How do I get this?

Answer 1

I assume your equations are

√(2)cosx-1=(1+√3)/2

√(2)cosx+1=(1-√3)/2  If this is so then multiply them together to get

2cos²x-1=(1-3)/4 or 2cos²x=1-2/4=1/2

cos²x=1/4 or case 1.  cosx=1/2 or x=π/3 4x=4π/3 whose cosine is -1/2

                    case 2.  cosx=-1/2 or x=2π/3 4x=8π/3 whose cosine is -1/2