Why do the solutions to the equation cos²x-3/4 lie in all four quadrants? Please explain why.

I am assuming that you mean cos^2 x - 3/4 because 3.4 is
not valid.

==> cos^2 x - 3/4 =
0

Let us solve the
equation.

First add 3/4 to both
sides.

==> cos^2 x =
3/4

Now we will take the root of both
sides.

==> cos(x) =
+-sqrt3/2

==> Then, we have two possible values for
cosx.

Case(1): cosx =+ sqrt3/2 ==> x > 0.
Then, x is in the first and fourth
quadrants.

==> x1= pi/6  ( first
quadrant)

==> x2= 2pi -
pi/6 = 11pi/6 ( fourth quadrant).

Case(2):
cosx = - sqrt3/2 ==> x < 0. Then, x is in the second and third
quadrants.

==> x3= pi- pi/6 = 5pi/6 (
2nd quadrant).

==> x4=
pi + pi/6 = 7pi/6 ( 3rd
quadrant).

Then, the solutions
of cos^2 x - 3/4 are in all four quadrants.