Let's solve the problem step by step.

Expression to Simplify:

$\text{Find the exact value of } \tan\left(\frac{4\pi}{3} + \frac{7\pi}{4}\right)$

We can apply the sum of angles identity for tangent: $\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)}$

For $A = \frac{4\pi}{3}$ and $B = \frac{7\pi}{4}$, let's first calculate the tangent values of each angle individually.

1. Calculate $\tan\left(\frac{4\pi}{3}\right)$:

$\frac{4\pi}{3}\text{ is in the third quadrant (where tangent is positive). }$ The reference angle is: $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$ Thus, $\tan\left(\frac{4\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}$

2. Calculate $\tan\left(\frac{7\pi}{4}\right)$:

$\frac{7\pi}{4}\text{ is in the fourth quadrant (where tangent is negative). }$ The reference angle is: $2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$ Thus, $\tan\left(\frac{7\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1$

3. Apply the sum of angles identity:

$\tan\left(\frac{4\pi}{3} + \frac{7\pi}{4}\right) = \frac{\tan\left(\frac{4\pi}{3}\right) + \tan\left(\frac{7\pi}{4}\right)}{1 - \tan\left(\frac{4\pi}{3}\right) \cdot \tan\left(\frac{7\pi}{4}\right)}$ Substitute the values: