Find cos4x in terms of cosx.

As with any question involving sines and cosines, consider complex numbers as a likely way to find the answer. For this particular kind of question, where sin/cos of a multiple of x is needed in terms of sinx/cosx, or vice versa, the complex number representation z = cosx + isinx must be expanded in two ways. 

Firstly, expand using de Moivre's Theorem:

z = cosx + isinx

z4 = (cosx + isinx)4 =  cos4x + sin4x

Then, expand using the binomial expansion formula to get powers of cosx:

z4 = (cosx + isinx)4 = cos4x + 4icos3xsinx + 6i2cos2xsin2x + 4i3cosxsin3x + i4sin4x

                               = cos4x + 4icos3xsinx - 6cos2xsin2x - 4icosxsin3x + sin4x

Equate the real parts of both expansions to get cos equivalence:

cos4x = cos4x - 6cos2xsin2x + sin4x

Use cos2x + sin2x = 1 as a substitution:

cos4x = cos4x - 6cos2x(1-cos2x) + (1-cos2x)2

          = 8cos4x - 8cos2x + 1

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