I think you are making this more difficult than it really is....

4cos4θ=0

the four is insignificant (because 0/4 is still 0). so now we have:

cos4θ=0.

I think the 4 is confusing you so let x=4θ so now we have:

cosx=0

and the solutions to that is when x=pi/2 or 3pi/2

now substitute 4θ in for x and you have

4θ=pi/2 and 4θ=3pi/2

so dividing by four yields:

θ=pi/8 and θ=3pi/8

Hope that helped!!