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Let's solve the given trigonometric identity:

We are tasked to prove that:

\[
\tan^2 \theta - \sin^2 \theta = \sin^4 \theta \sec^2 \theta
\]

### Step 1: Express \(\tan^2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)
We know that:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
So,
\[
\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}
\]

### Step 2: Substitute \(\tan^2 \theta\) into the left-hand side of the equation
The left-hand side becomes:
\[
\frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta
\]

### Step 3: Combine the terms over a common denominator
Rewrite \(\sin^2 \theta\) as \(\frac{\sin^2 \theta \cdot \cos^2 \theta}{\cos^2 \theta}\) to have a common denominator:
\[
\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cdot \cos^2 \theta}{\cos^2 \theta} = \frac{\sin^2 \theta - \sin^2 \theta \cdot \cos^2 \theta}{\cos^2 \theta}
\]

### Step 4: Factor the numerator
Factor out \(\sin^2 \theta\) from the numerator:
\[
\frac{\sin^2 \theta (1 - \cos^2 \theta)}{\cos^2 \theta}
\]

### Step 5: Use the Pythagorean identity
We know that:
\[
1 - \cos^2 \theta = \sin^2 \theta
\]
Substitute this into the expression:
\[
\frac{\sin^2 \theta \cdot \sin^2 \theta}{\cos^2 \theta} = \frac{\sin^4 \theta}{\cos^2 \theta}
\]

### Step 6: Express the result in terms of \(\sec^2 \theta\)
Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\).
Thus, the right-hand side becomes:
\[
\sin^4 \theta \sec^2 \theta
\]

This matches the right-hand side of the equation, so the identity is proven:

\[
\tan^2 \theta - \sin^2 \theta = \sin^4 \theta \sec^2 \theta
\]

---

Would you like further details on any step?

Here are 5 related questions to deepen your understanding:
1. How does the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) assist in solving trigonometric proofs?
2. Can you derive other identities using the basic relationships between \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\)?
3. What is the significance of factoring in simplifying trigonometric expressions?
4. How does the secant function relate to \(\cos \theta\) and what are its practical applications?
5. Can you prove \(\sec^2 \theta - \tan^2 \theta = 1\) using similar techniques?

**Tip**: When proving trigonometric identities, always try to rewrite everything in terms of \(\sin \theta\) and \(\cos \theta\) for consistency and easier manipulation.

Show that tan²θ - sin²θ = sin⁴θ sec²θ.

Step-by-step solution to proving the trigonometric identity tan²θ - sin²θ = sin⁴θ sec²θ. This involves using trigonometric formulas like tan(θ) = sin(θ)/cos(θ) and applying the Pythagorean identity. Suitable for high school students in Grades 10-12.

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