Final answer:
The trigonometric expression tan4x - sec4x simplifies to -2tan2x - 1 after recognizing the difference of squares and using a trigonometric identity for sec2x.
Step-by-step explanation:
To factor the trigonometric expression tan4x - sec4x, we start by recognizing a difference of squares. The expression can be written as:
(tan2x + sec2x)(tan2x - sec2x)
However, we know that sec2x = 1 + tan2x, which we can substitute into our expression.
So by substituting, our expression becomes:
(tan2x + (1 + tan2x))(tan2x - (1 + tan2x))
Simplified further, it results in:
(2tan2x + 1)(-1)
Finally, the expression simplifies to -2tan2x - 1, which is represented by option B.