Use the sum to product formula to prove that cos 2 theta - sin 2 theta = square root 2, can be rewritten as sin( 2theta - pie/4) = -1.

To prove that \(\cos 2\theta - \sin 2\theta = \sqrt{2}\) can be rewritten as \(\sin(2\theta - \frac{\pi}{4}) = -1\), we will use trigonometric identities and transformations. Let's break it down step-by-step.

### Step 1: Using Sum to Product Formula
Recall the sum to product identities for sine and cosine:
\[
\cos A - \sin A = \sqrt{2} \cos\left(A + \frac{\pi}{4}\right)
\]

For \(A = 2\theta\), this becomes:
\[
\cos 2\theta - \sin 2\theta = \sqrt{2} \cos\left(2\theta + \frac{\pi}{4}\right)
\]

Given that:
\[
\cos 2\theta - \sin 2\theta = \sqrt{2}
\]

We substitute the sum to product identity:
\[
\sqrt{2} \cos\left(2\theta + \frac{\pi}{4}\right) = \sqrt{2}
\]

### Step 2: Simplifying the Equation
Divide both sides by \(\sqrt{2}\):
\[
\cos\left(2\theta + \frac{\pi}{4}\right) = 1
\]

### Step 3: Solving for the Argument of the Cosine
The cosine function equals 1 at specific angles. Specifically:
\[
\cos x = 1 \quad \text{when} \quad x = 2k\pi \quad (k \in \mathbb{Z})
\]

Therefore, we have:
\[
2\theta + \frac{\pi}{4} = 2k\pi \quad \text{for some integer} \quad k
\]

### Step 4: Isolating \(\theta\)
Solve for \(\theta\):
\[
2\theta = 2k\pi - \frac{\pi}{4}
\]

Divide both sides by 2:
\[
\theta = k\pi - \frac{\pi}{8}
\]

### Step 5: Verifying the Identity with Sine
Now we consider the trigonometric identity we need to prove:
\[
\sin\left(2\theta - \frac{\pi}{4}\right) = -1
\]

Substitute \(\theta = k\pi - \frac{\pi}{8}\) into \(2\theta\):
\[
2\theta = 2(k\pi - \frac{\pi}{8}) = 2k\pi - \frac{\pi}{4}
\]

Therefore,
\[
2\theta - \frac{\pi}{4} = (2k\pi - \frac{\pi}{4}) - \frac{\pi}{4} = 2k\pi - \frac{\pi}{2}
\]

Recall that:
\[
\sin x = -1 \quad \text{when} \quad x = \frac{3\pi}{2} + 2n\pi \quad (n \in \mathbb{Z})
\]

Thus, for our expression:
\[
2k\pi - \frac{\pi}{2} = \frac{3\pi}{2} + 2n\pi
\]

For some integer \(n\), this simplifies to:
\[
2k\pi - \frac{\pi}{2} = \frac{3\pi}{2} + 2n\pi
\]

Therefore, the expression holds true. Hence, we have shown:
\[
\cos 2\theta - \sin 2\theta = \sqrt{2} \implies \sin\left(2\theta - \frac{\pi}{4}\right) = -1
\]

Would you like more details or have any questions?

Here are 5 questions you might ask next:
1. How do you derive the sum to product formulas for sine and cosine?
2. Can you provide more examples of using the sum to product formulas?
3. What are the specific angles where sine and cosine take on their extreme values?
4. How do you prove other trigonometric identities using these transformations?
5. How can these identities be applied in solving real-world problems?

**Tip:** When solving trigonometric identities, always look for opportunities to use known formulas and identities to transform the given expressions into simpler forms.

Learn how to prove that cos 2 theta - sin 2 theta = sqrt(2) using trigonometric identities and transformations. This problem involves applying the sum to product formulas for sine and cosine, demonstrating advanced trigonometric principles.

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