The question was Find the roots of z^4 + 1 = 0 and show them in an argand diagram. Resolve z^4 + 1 into real quadratic factors and deduce that cos2a = 2(cosa - cospi/4)(cosa - cos3pi/4)
z^4 = -1
= (cospi + i * sinpi) or to shorten that cispi
z = cis((pi + 2pi * k) / 4) where k = 0, 1, ... 3
z = cispi/4, cis3pi/4, cis5pi/4, cis7pi/4
for later work cispi/4 = to the conjugate of cis7pi/4
and cis3pi/4 = to the conjugate of cis5pi/4
Drew argand. But while we could prove the above statement, we couldn't deduce it from the roots of z^4 + 1

It's always a hard task to second-guess somebody's meaning. What might "deduce it from the roots of z^4 + 1" mean? The fact that cos(p/4) is a real part of a root of z⁴ + 1 = 0 (as is cos(3p/4))in conjunction with the Argand diagram suggests a value for cos(p/4). Once you know that value, the statement is deduced easily.

I think you should approach your teacher with the question about the meaning of that "deduction" request.