To find which expression is equivalent to \(\tan \left(\frac{3 \pi}{4} - 2x\right)\), we can use the tangent subtraction formula:
[tex]\[
\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)}
\][/tex]
Here, let \(a = \frac{3 \pi}{4}\) and \(b = 2x\).
First, we need to determine \(\tan\left(\frac{3 \pi}{4}\right)\). Knowing the tangent values of common angles, we have:
[tex]\[
\tan\left(\frac{3 \pi}{4}\right) = \tan\left(135^\circ\right) = -1
\][/tex]
Using the tangent subtraction formula:
[tex]\[
\tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{\tan\left(\frac{3 \pi}{4}\right) - \tan(2x)}{1 + \tan\left(\frac{3 \pi}{4}\right) \tan(2x)}
\][/tex]
Substitute \(\tan\left(\frac{3 \pi}{4}\right) = -1\):
[tex]\[
\tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{-1 - \tan(2x)}{1 + (-1) \cdot \tan(2x)}
\][/tex]
Simplify the denominator:
[tex]\[
\tan\left(\frac{3 \pi}{4} - 2x\right) = \frac{-1 - \tan(2x)}{1 - \tan(2x)}
\][/tex]
Thus, the expression equivalent to \(\tan\left(\frac{3 \pi}{4} - 2x\right)\) is:
[tex]\[
\frac{-1 - \tan(2x)}{1 - \tan(2x)}
\][/tex]
So, the correct choice is the first option:
[tex]\[
\boxed{\frac{-1 - \tan(2x)}{1 - \tan(2x)}}
\][/tex]
