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The given identity to prove is:

\[
\frac{\sin^2 \theta - 3\cos^2 \theta + 1}{\sin^2 \theta - \cos^2 \theta} \equiv 2
\]

We also need to identify any values of \(\theta\) for which the identity is not valid.

### Step-by-step proof:

1. **Simplifying the numerator**:
   \[
   \text{Numerator: } \sin^2 \theta - 3\cos^2 \theta + 1
   \]
   Using the Pythagorean identity, \(\sin^2 \theta + \cos^2 \theta = 1\), we substitute for \(\sin^2 \theta\):
   \[
   \sin^2 \theta = 1 - \cos^2 \theta
   \]
   So the numerator becomes:
   \[
   (1 - \cos^2 \theta) - 3\cos^2 \theta + 1 = 2 - 4\cos^2 \theta
   \]
   
2. **Simplifying the denominator**:
   \[
   \text{Denominator: } \sin^2 \theta - \cos^2 \theta
   \]
   Again, using \(\sin^2 \theta = 1 - \cos^2 \theta\), the denominator becomes:
   \[
   (1 - \cos^2 \theta) - \cos^2 \theta = 1 - 2\cos^2 \theta
   \]

3. **Rewriting the expression**:
   Now the expression becomes:
   \[
   \frac{2 - 4\cos^2 \theta}{1 - 2\cos^2 \theta}
   \]

4. **Checking the identity**:
   Let's divide the numerator and the denominator by \((1 - 2\cos^2 \theta)\):
   \[
   \frac{2 - 4\cos^2 \theta}{1 - 2\cos^2 \theta} = 2
   \]
   This confirms the given identity.

### Values of \(\theta\) for which the identity is not valid:

The identity will not be valid if the denominator becomes zero, i.e., if:

\[
1 - 2\cos^2 \theta = 0
\]

Solving for \(\cos^2 \theta\):

\[
\cos^2 \theta = \frac{1}{2}
\]

Therefore, \(\cos \theta = \pm \frac{1}{\sqrt{2}}\), which corresponds to:

\[
\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \dots
\]

Thus, the identity is not valid for \(\theta = \frac{\pi}{4} + n\pi\), where \(n \in \mathbb{Z}\).

Would you like more details on this proof or help with related concepts?

Here are 5 relative questions for deeper understanding:
1. What is the significance of trigonometric identities in simplifying expressions?
2. How does the Pythagorean identity help in transforming trigonometric expressions?
3. What methods can be used to verify trigonometric identities?
4. How can you determine when a trigonometric identity is undefined?
5. What other common angles cause trigonometric expressions to be undefined?

**Tip**: Always check for points where the denominator becomes zero when dealing with trigonometric identities to avoid undefined expressions.

Prove each of the following identities. In each case, identify any values of θ for which the identity is not valid. (i) (sin^2(θ) - 3cos^2(θ) + 1) / (sin^2(θ) - cos^2(θ)) ≡ 2

Step-by-step proof for the trigonometric identity (sin^2(θ) - 3cos^2(θ) + 1) / (sin^2(θ) - cos^2(θ)) ≡ 2. Identify values of θ where the identity is not valid. Learn how to use the Pythagorean identity to simplify expressions.

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