Use a half angle formula or formula for reducing powers to fill in the blanks in the identity below: ( cos ( 4 x ) )^2 = ___________ + ____________cos(____x)

Answer 1

cos(a/2) = √((1+cos(a))/2) is how the half angle formula is normally expressed. Squaring both sides,

cos²(a/2) = (1 + cos(a))/2 = 1/2 + 1/2 cos(a)

Let a = 8x

Then, cos²(8x/2) = 1/2 + 1/2 cos(8x) or

cos²(4x) = 1/2 + 1/2 cos(8x)