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To solve the given limit using trigonometric identities, the expression is:

\[
\lim_{{x \to \frac{\pi}{4}}} \left( \frac{\cos(2x)}{\sin(x) - \cos(x)} + \tan(x) + 1 \right)
\]

Let's break this down step by step using trigonometric identities.

### Step 1: Trigonometric identities
1. **Double-angle identity for cosine:**
   \[
   \cos(2x) = \cos^2(x) - \sin^2(x)
   \]
2. **Simplify the term \(\tan(x)\):**
   \[
   \tan(x) = \frac{\sin(x)}{\cos(x)}
   \]

### Step 2: Substitute \(\cos(2x)\) and simplify
The expression becomes:

\[
\lim_{{x \to \frac{\pi}{4}}} \left( \frac{\cos^2(x) - \sin^2(x)}{\sin(x) - \cos(x)} + \frac{\sin(x)}{\cos(x)} + 1 \right)
\]

### Step 3: Evaluate at \(x = \frac{\pi}{4}\)
At \(x = \frac{\pi}{4}\), the values of the basic trigonometric functions are:

- \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
- \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Substitute these values into the limit expression:

1. **Cosine double angle at \(\frac{\pi}{4}\):**
   \[
   \cos(2x) = \cos\left(\frac{\pi}{2}\right) = 0
   \]
   So the first term becomes:
   \[
   \frac{\cos^2\left(\frac{\pi}{4}\right) - \sin^2\left(\frac{\pi}{4}\right)}{\sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right)} = \frac{0}{0}
   \]
   This is an indeterminate form, which suggests we need to simplify further.

### Step 4: Apply L'Hopital's Rule
Since the first term gives an indeterminate form \(\frac{0}{0}\), we apply L'Hopital's Rule by differentiating the numerator and denominator:

- **Derivative of the numerator \(\cos(2x)\):**
  \[
  \frac{d}{dx}[\cos(2x)] = -2\sin(2x)
  \]
- **Derivative of the denominator \(\sin(x) - \cos(x)\):**
  \[
  \frac{d}{dx}[\sin(x) - \cos(x)] = \cos(x) + \sin(x)
  \]

So the limit of the first term becomes:

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

At \(x = \frac{\pi}{4}\):

- \(\sin\left(\frac{\pi}{2

Solve the limit as x approaches π/4 of (cos(2x)/(sin(x) - cos(x)) + tan(x) + 1).

Solve the limit of (cos(2x)/(sin(x) - cos(x)) + tan(x) + 1) as x approaches π/4 using trigonometric identities and L'Hopital's Rule. Detailed step-by-step explanation provided for Grades 11-12 students.

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