The problem asks us to find the interval where the function $f(x) = \cos(\frac{\pi}{4} - x)$ is monotonically increasing.

1. Problem Description

The problem asks us to find the interval where the function $f(x) = \cos(\frac{\pi}{4} - x)$ is monotonically increasing.

2. Solution Steps

First, we can rewrite the cosine function using the identity $\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$:

$f(x) = \cos(\frac{\pi}{4} - x) = \sin(\frac{\pi}{2} - (\frac{\pi}{4} - x)) = \sin(\frac{\pi}{2} - \frac{\pi}{4} + x) = \sin(\frac{\pi}{4} + x)$

The sine function $\sin(x)$ is increasing in the interval $[-\frac{\pi}{2} + 2k\pi, \frac{\pi}{2} + 2k\pi]$, where $k \in \mathbb{Z}$.

Therefore, the function $\sin(\frac{\pi}{4} + x)$ is increasing when:

$-\frac{\pi}{2} + 2k\pi \le \frac{\pi}{4} + x \le \frac{\pi}{2} + 2k\pi$

Subtracting $\frac{\pi}{4}$ from all parts of the inequality, we get:

$-\frac{\pi}{2} - \frac{\pi}{4} + 2k\pi \le x \le \frac{\pi}{2} - \frac{\pi}{4} + 2k\pi$

$-\frac{3\pi}{4} + 2k\pi \le x \le \frac{\pi}{4} + 2k\pi$

So, the increasing interval is $[-\frac{3\pi}{4} + 2k\pi, \frac{\pi}{4} + 2k\pi]$, where $k \in \mathbb{Z}$.

We can also use the fact that $\cos(x)$ is decreasing on $[0, \pi]$ and increasing on $[\pi, 2\pi]$.

So $f(x) = \cos(\frac{\pi}{4} - x)$ is increasing when $\pi + 2k\pi \le \frac{\pi}{4} - x \le 2\pi + 2k\pi$, $k \in \mathbb{Z}$

Multiplying by -1, we have

$-2\pi - 2k\pi \le x - \frac{\pi}{4} \le -\pi - 2k\pi$

Add $\frac{\pi}{4}$ to all parts

$-2\pi + \frac{\pi}{4} - 2k\pi \le x \le -\pi + \frac{\pi}{4} - 2k\pi$

$-\frac{7\pi}{4} - 2k\pi \le x \le -\frac{3\pi}{4} - 2k\pi$

Multiply by -1, we have

$\frac{3\pi}{4} + 2k\pi \le x \le \frac{7\pi}{4} + 2k\pi$, which is incorrect.

Then use the increasing interval for $\cos x$:

$(2k+1)\pi \le \frac{\pi}{4} - x \le (2k+2)\pi$.

$-\frac{7\pi}{4} - 2k\pi \le x \le -\frac{3\pi}{4} - 2k\pi$

$\frac{3\pi}{4} + 2k\pi \le -x \le \frac{7\pi}{4} + 2k\pi$.

Therefore, the interval is $[\frac{3\pi}{4} + 2k\pi, \frac{7\pi}{4} + 2k\pi]$.

3. Final Answer

The increasing interval is $[-\frac{3\pi}{4} + 2k\pi, \frac{\pi}{4} + 2k\pi]$, $k \in \mathbb{Z}$.

Final Answer: The final answer is $\boxed{[-\frac{3\pi}{4} + 2k\pi, \frac{\pi}{4} + 2k\pi], k \in Z}$