To rewrite the expression -1 + tan(x) / (1 + tan(x)), we can start by finding a common denominator for the terms in the numerator.
Multiplying the numerator and denominator by 1 - tan(x), we get:
((-1 + tan(x)) * (1 - tan(x))) / ((1 + tan(x)) * (1 - tan(x)))
Expanding the numerator, we have:
(-1 + tan(x) - tan(x) + tan^2(x)) / (1 - tan^2(x))
Simplifying, we get:
(tan^2(x) - 2tan(x) - 1) / (-tan^2(x) + 1)
Now, we can factor the numerator and denominator:
[(tan(x) - 1)(tan(x) + 1)] / [(1 - tan(x))(1 + tan(x))]
Canceling out the common factors, we have:
-(tan(x) - 1) / (1 - tan(x))
Rearranging, we get:
(1 - tan(x)) / (tan(x) - 1)
Therefore, the expression -1 + tan(x) / (1 + tan(x)) can be rewritten as:
(1 - tan(x)) / (tan(x) - 1)
Comparing this expression to the answer choices given, none of the options A, B, C, or D matches the rewritten expression.