Write the following as a single sine or cosine function of an angle: \( \cos ^{4} 4 x-\sin ^{4} 4 x \)

To simplify the expression \( \cos^{4}(4x) - \sin^{4}(4x) \), we can use the difference of squares formula: \[ a^2 - b^2 = (a - b)(a + b) \] In this case, let \( a = \cos^2(4x) \) and \( b = \sin^2(4x) \). Then we have: \[ \cos^{4}(4x) - \sin^{4}(4x) = (\cos^{2}(4x) - \sin^{2}(4x))(\cos^{2}(4x) + \sin^{2}(4x)) \] Since \( \cos^{2}(4x) + \sin^{2}(4x) = 1 \), the expression simplifies to: \[ \cos^{4}(4x) - \sin^{4}(4x) = \cos^{2}(4x) - \sin^{2}(4x) \] Next, we can use the double angle identity: \[ \cos^{2}(A) - \sin^{2}(A) = \cos(2A) \] Applying this to our expression: \[ \cos^{2}(4x) - \sin^{2}(4x) = \cos(2 \cdot 4x) = \cos(8x) \] Thus, the final result is: \[ \cos^{4}(4x) - \sin^{4}(4x) = \cos(8x) \]