Step-by-step Solution:

\[ \tan\left(\frac{\pi}{4} + \theta\right) \cdot \tan\left(\frac{3\pi}{4} + \theta\right). \] Step 1: Write the formulas for \(\tan(a + b)\) \[\] For \(\tan\left(\frac{\pi}{4} + \theta\right)\): \[\] Using the formula \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\), we know: \[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan\theta}{1 - \tan\theta}. \] For \(\tan\left(\frac{3\pi}{4} + \theta\right)\): \[\] Since \(\tan\frac{3\pi}{4} = -1\), we have: \[ \tan\left(\frac{3\pi}{4} + \theta\right) = \frac{-1 + \tan\theta}{1 + \tan\theta}. \] Step 2: Now, multiply \(\tan\left(\frac{\pi}{4} + \theta\right)\) and \(\tan\left(\frac{3\pi}{4} + \theta\right)\): \[ \tan\left(\frac{\pi}{4} + \theta\right) \cdot \tan\left(\frac{3\pi}{4} + \theta\right) = \frac{1 + \tan\theta}{1 - \tan\theta} \cdot \frac{-1 + \tan\theta}{1 + \tan\theta}. \] Step 3: Simplify The numerator and denominator simplify as follows: \[ \tan\left(\frac{\pi}{4} + \theta\right) \cdot \tan\left(\frac{3\pi}{4} + \theta\right) = \frac{(1 + \tan\theta)(-1 + \tan\theta)}{(1 - \tan\theta)(1 + \tan\theta)}. \] \[ = {-1}\]