Final answer:
The trigonometric identity for the cosine of double angles, cos(2θ) = cos²(θ) - sin²(θ), indicates that 'b' is 8x. However, this answer is not listed in the options, suggesting an error in the question or answer choices.
Step-by-step explanation:
To solve the mathematical problem where we have cos²(4x) - sin²(4x) = cos(b), we can employ a trigonometric identity for the cosine of double angles.
The identity states cos(2θ) = cos²(θ) - sin²(θ). By comparing this identity with our equation, we can see that b must be twice the angle of 4x, which means b = 2(4x) or b = 8x.
However, since none of the answer options matches 8x, we might consider that there is an error in the question or answer choices.
Nonetheless, if we continue to evaluate the options provided based on the identity and the information given, the option that closely resembles the double-angle identity is option A) 4x, despite it not being the exact solution.
The answer options do not correctly represent the result of the trigonometric identity applied to the equation. This indicates a potential error in the question formulation or answer choices.