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To solve the identity:

\[
\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \times \sin^2 \theta
\]

we will attempt to transform the left-hand side (LHS) and the right-hand side (RHS) of the equation to prove that both expressions are equivalent.

### Step 1: Express \(\tan^2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)

We know the identity for \(\tan \theta\):

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

So, \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\).

### Step 2: Substitute \(\tan^2 \theta\) in both sides of the equation

The equation becomes:

\[
\frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \times \sin^2 \theta
\]

### Step 3: Simplify both sides

#### Left-hand side (LHS):
\[
\frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta
\]
We can rewrite this as:
\[
\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta}
\]
\[
= \frac{\sin^2 \theta - \sin^2 \theta \cos^2 \theta}{\cos^2 \theta}
\]
\[
= \frac{\sin^2 \theta(1 - \cos^2 \theta)}{\cos^2 \theta}
\]

Now, using the Pythagorean identity \(1 - \cos^2 \theta = \sin^2 \theta\), we get:
\[
\frac{\sin^2 \theta \times \sin^2 \theta}{\cos^2 \theta}
\]
\[
= \frac{\sin^4 \theta}{\cos^2 \theta}
\]

#### Right-hand side (RHS):
The right-hand side is:
\[
\frac{\sin^2 \theta}{\cos^2 \theta} \times \sin^2 \theta
\]
\[
= \frac{\sin^4 \theta}{\cos^2 \theta}
\]

### Step 4: Conclude the proof

Since both sides are equal:

\[
\frac{\sin^4 \theta}{\cos^2 \theta} = \frac{\sin^4 \theta}{\cos^2 \theta}
\]

The given equation is true, and thus:

\[
\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \times \sin^2 \theta
\]

Would you like further clarification or additional details?

Here are some related questions:
1. What are some other common trigonometric identities?
2. How can the Pythagorean identity be applied in other contexts?
3. What happens when \(\theta = 0\) or \(\theta = \frac{\pi}{2}\)?
4. How can the double-angle identities help in simplifying trigonometric expressions?
5. What is the geometric interpretation of \(\tan \theta\) and \(\sin \theta\) on the unit circle?

**Tip**: When proving trigonometric identities, converting all terms to \(\sin \theta\) and \(\cos \theta\) often simplifies the process.

Show that tan²θ - sin²θ = tan²θ × sin²θ

In this trigonometry problem, we prove the identity tan²θ - sin²θ = tan²θ × sin²θ. The solution includes key concepts like the tangent and sine functions, trigonometric identities, and the Pythagorean identity. Suitable for students in Grades 10-12.

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