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

To simplify the expression, we can manipulate the trigonometric functions involved. Starting with the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can rewrite the fraction involving \( \sin^2 4x \). Then, using the identity \( 1 + \cos 4x = 2 \cos^2 2x \), we can further simplify it. So, the expression becomes: \( = 1 + \frac{1 - \cos^2 4x}{(1 + \cos 4x)^2} = 1 + \frac{1 - \cos^2 4x}{(2 \cos^2 2x)^2} \) After proper substitutions and canceling common terms, you will find that the expression simplifies down nicely, perhaps leading to a cleaner form depending on further algebraic manipulation. If you plug in specific values for \( x \), you can confirm your findings and gain more insights into how this function behaves across different scenarios, making it easier to visualize the output and its application!