\( = \frac { 1 } { 1 } + \frac { \sin ^ { 2 } 4 x } { ( 1 + \cos 4 x ) ^ { 2 } } = \)

Given the expression \[ 1 + \frac{\sin^2 4x}{(1+\cos4x)^2}, \] we start by rewriting parts of it using trigonometric identities. 1. Recall the double-angle formulas: \[ 1+\cos4x = 2\cos^2 2x \] and \[ \sin4x = 2\sin2x\cos2x. \] 2. Substitute these into the fraction: \[ \frac{\sin^2 4x}{(1+\cos4x)^2} = \frac{(2\sin2x\cos2x)^2}{(2\cos^2 2x)^2}. \] 3. Square the numerator and denominator: \[ \frac{(2\sin2x\cos2x)^2}{(2\cos^2 2x)^2} = \frac{4\sin^2 2x\cos^2 2x}{4\cos^4 2x}. \] 4. Cancel out the common factor of 4: \[ \frac{4\sin^2 2x\cos^2 2x}{4\cos^4 2x} = \frac{\sin^2 2x\cos^2 2x}{\cos^4 2x} = \frac{\sin^2 2x}{\cos^2 2x} = \tan^2 2x. \] 5. Substitute back into the original expression: \[ 1 + \frac{\sin^2 4x}{(1+\cos4x)^2} = 1 + \tan^2 2x. \] 6. Use the Pythagorean identity: \[ 1 + \tan^2 \theta = \sec^2 \theta, \] with \(\theta = 2x\), so: \[ 1 + \tan^2 2x = \sec^2 2x. \] Thus, the simplified expression is: \[ \sec^2 2x. \]