This is how I did it:

z^4 + 1 = (z^2 - 2zcos (pi/4) + 1)(z^2 -2zcos (3pi/4) + 1)

Divide both sides by z^2 (for RHS, I just divided each bracket by z, which is the equivalent of dividing by z^2):

z^2 + 1/z^2 = (z - 2cos (pi/4) + 1/z)(z - 2cos (3pi/4) + 1/z)

Now group z + 1/z and change them to a value of cos@ (z + 1/z = 2cos@ and z^2 + 1/z^2 = 2cos2@):

Therefore; 2cos2@ = (2cos@ - 2cos (pi/4))(2cos@ - 2cos (3pi/4))

2cos2@ = 4 (cos@ - cos (pi/4))(cos@ - cos (3pi/4))

Hence, cos2@ = 2 (cos@ - cos (pi/4))(cos@ - cos (3pi/4))

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