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To prove the given trigonometric identity:

\[
\frac{\cos 2x}{(\cos x + \sin x)^2} = \frac{\cos x - \sin x}{1 + \sin 2x}
\]

### Step 1: Simplify the left-hand side (LHS)

Starting with the left-hand side:

\[
\frac{\cos 2x}{(\cos x + \sin x)^2}
\]

First, recall the double angle identity for cosine:

\[
\cos 2x = \cos^2 x - \sin^2 x
\]

Now, we need to expand the denominator:

\[
(\cos x + \sin x)^2 = \cos^2 x + 2\cos x \sin x + \sin^2 x
\]

Using the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\), we get:

\[
(\cos x + \sin x)^2 = 1 + 2\cos x \sin x
\]

This can also be written as:

\[
(\cos x + \sin x)^2 = 1 + \sin 2x
\]

Thus, the left-hand side becomes:

\[
\frac{\cos 2x}{1 + \sin 2x}
\]

### Step 2: Simplify the right-hand side (RHS)

Now, let's look at the right-hand side:

\[
\frac{\cos x - \sin x}{1 + \sin 2x}
\]

### Step 3: Compare the left-hand side with the right-hand side

At this point, both sides have the same denominator, \(1 + \sin 2x\). So, for the identity to hold true, the numerators must be equal.

However, from our previous steps, we have the numerator on the left-hand side as \(\cos 2x\) and on the right-hand side as \(\cos x - \sin x\).

The identity that can relate these two expressions is:

\[
\cos 2x = (\cos x - \sin x)(\cos x + \sin x)
\]

Expanding the right-hand side:

\[
(\cos x - \sin x)(\cos x + \sin x) = \cos^2 x - \sin^2 x = \cos 2x
\]

This confirms that:

\[
\frac{\cos 2x}{1 + \sin 2x} = \frac{\cos x - \sin x}{1 + \sin 2x}
\]

Hence, the identity is proven.

---

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How can the Pythagorean identity be used to simplify trigonometric expressions?
2. What are the common trigonometric identities involving double angles?
3. How do you expand binomials in trigonometry?
4. What is the significance of the identity \(\cos^2 x + \sin^2 x = 1\)?
5. How can you derive the sum-to-product identities from the basic trigonometric identities?

**Tip:** Always check if the identities on both sides can be transformed into a common form, especially focusing on the numerators and denominators separately.

Learn how to prove the trigonometric identity Cos 2x / (Cos x + Sin x)^2 = (Cos x - Sin x) / (1 + Sin 2x) step-by-step with detailed explanations and applications of double angle identities and the Pythagorean identity.

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