5.2.2 For which values of x in the interval [0 ° ; 360° ] is the identity undefined?

To determine the values of \( x \) in the interval \([0^\circ; 360^\circ]\) where the given trigonometric identity is undefined, we must examine where the denominator of either side of the identity is equal to zero.

### Identity Recap:
The identity is:

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

### Step 1: Denominator Conditions

1. **Denominator on the left-hand side (LHS):**
   \[
   (\cos x + \sin x)^2
   \]
   This expression will be zero when \(\cos x + \sin x = 0\).

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

   Dividing both sides by \(\cos x\) (assuming \(\cos x \neq 0\)):

   \[
   1 = -\tan x
   \]

   Therefore:

   \[
   \tan x = -1
   \]

   \(\tan x = -1\) occurs at:
   \[
   x = 135^\circ, 315^\circ \quad \text{(within the interval [0°, 360°])}
   \]

2. **Denominator on the right-hand side (RHS):**
   \[
   1 + \sin 2x
   \]
   This expression will be zero when:

   \[
   1 + \sin 2x = 0
   \]
   \[
   \sin 2x = -1
   \]

   \(\sin 2x = -1\) occurs when:
   \[
   2x = 270^\circ + 360^\circ n, \quad \text{for integer } n
   \]

   Simplifying for \( x \):
   \[
   x = 135^\circ + 180^\circ n
   \]

   The relevant values of \( x \) in \([0^\circ; 360^\circ]\) are:
   \[
   x = 135^\circ, 315^\circ
   \]

### Step 2: Identify Undefined Values
From both the LHS and RHS, the values of \( x \) that make the identity undefined are:

\[
x = 135^\circ, 315^\circ
\]

These are the points where the identity is undefined within the interval \([0^\circ; 360^\circ]\).

---

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How can we find the values of \( x \) where \(\sin x = -1\) within a given interval?
2. What is the significance of the periodicity of trigonometric functions in solving equations?
3. Why is division by zero undefined in trigonometric identities?
4. How do changes in angles like \(2x\) affect the solutions in trigonometric equations?
5. How can the identity \(\tan x = -1\) be solved for other intervals?

**Tip:** Always check the domain and possible restrictions when dealing with trigonometric identities, especially where the denominator could be zero.

Discover where the trigonometric identity \(\frac{\cos 2x}{(\cos x + \sin x)^2} = \frac{\cos x - \sin x}{1 + \sin 2x}\) is undefined within the interval [0°; 360°]. Understand the conditions for undefined values based on the denominator of both sides of the identity. Suitable for high school students studying trigonometry.

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