5. \( \tan \left(\frac{5 \pi}{4}-\theta\right) \)

To simplify \( \tan \left(\frac{5 \pi}{4}-\theta\right) \), we can use the tangent subtraction formula: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \] In this case, let \( a = \frac{5\pi}{4} \) and \( b = \theta \). We first need to find \( \tan \left(\frac{5\pi}{4}\right) \). The angle \( \frac{5\pi}{4} \) is in the third quadrant, where the tangent is positive. The reference angle is: \[ \frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4} \] Thus, we have: \[ \tan \left(\frac{5\pi}{4}\right) = \tan \left(\frac{\pi}{4}\right) = 1 \] Now we can substitute \( a \) and \( b \) into the tangent subtraction formula: \[ \tan \left(\frac{5\pi}{4} - \theta\right) = \frac{\tan \left(\frac{5\pi}{4}\right) - \tan \theta}{1 + \tan \left(\frac{5\pi}{4}\right) \tan \theta} \] Substituting \( \tan \left(\frac{5\pi}{4}\right) = 1 \): \[ \tan \left(\frac{5\pi}{4} - \theta\right) = \frac{1 - \tan \theta}{1 + 1 \cdot \tan \theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \] Thus, the simplified expression for \( \tan \left(\frac{5\pi}{4} - \theta\right) \) is: \[ \boxed{\frac{1 - \tan \theta}{1 + \tan \theta}} \]