Evaluate: `lim_(x to (pi)/4)(cos7x + cos5x)/(sin7x - sin5x)`

Answer

To evaluate the limit

\[ \lim_{x \to \frac{\pi}{4}} \frac{\cos(7x) + \cos(5x)}{\sin(7x) - \sin(5x)}, \]

we can use trigonometric identities to simplify the expression.

### Step 1: Apply the cosine addition identity

We know that

\[ \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right). \]

Using this identity for \(\cos(7x) + \cos(5x)\):

- Let \(a = 7x\) and \(b = 5x\).
- Then, \(a + b = 12x\) and \(a - b = 2x\).

Thus,

\[ \cos(7x) + \cos(5x) = 2 \cos\left(6x\right) \cos\left(x\right). \]

### Step 2: Apply the sine subtraction identity

We also know that

\[ \sin a - \sin b = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right). \]

Using this identity for \(\sin(7x) - \sin(5x)\):

- Again, let \(a = 7x\) and \(b = 5x\).
- Then, \(a + b = 12x\) and \(a - b = 2x\).

Thus,

\[ \sin(7x) - \sin(5x) = 2 \cos\left(6x\right) \sin\left(x\right). \]

### Step 3: Substitute back into the limit

Now substituting these results back into the limit, we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{2 \cos(6x) \cos(x)}{2 \cos(6x) \sin(x)}. \]

The \(2 \cos(6x)\) terms cancel out (as long as \(\cos(6x) \neq 0\), which is valid in the limit as \(x\) approaches \(\frac{\pi}{4}\)):

\[ = \lim_{x \to \frac{\pi}{4}} \frac{\cos(x)}{\sin(x)} = \lim_{x \to \frac{\pi}{4}} \cot(x). \]

### Step 4: Evaluate the limit

Now, we can evaluate the limit:

\[ \cot\left(\frac{\pi}{4}\right) = 1. \]

### Final Answer

Thus, the limit evaluates to:

\[ \boxed{1}. \]