Let's start by simplifying and analyzing the given trigonometric equation:

The equation provided is:

\[
\tan^2(x) - \sin^2(x) = \tan^2(x) \times \sin^2(x)
\]

### Step 1: Substitute \(\tan(x)\)
We know that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). So, \(\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}\).

Substituting \(\tan^2(x)\) into the equation:

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

### Step 2: Simplify the Left Side
Let's simplify the left side of the equation:

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

Factoring \(\sin^2(x)\) from the numerator:

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

We know that \(1 - \cos^2(x) = \sin^2(x)\). Substituting this, we get:

\[
\frac{\sin^4(x)}{\cos^2(x)}
\]

### Step 3: Simplify the Right Side
Now, simplify the right side of the equation:

\[
\frac{\sin^2(x)}{\cos^2(x)} \times \sin^2(x) = \frac{\sin^4(x)}{\cos^2(x)}
\]

### Step 4: Compare Both Sides
The simplified form of both the left and right sides is:

\[
\frac{\sin^4(x)}{\cos^2(x)} = \frac{\sin^4(x)}{\cos^2(x)}
\]

This shows that both sides of the equation are equal, confirming that the equation is **true**.

### Conclusion
The equation \(\tan^2(x) - \sin^2(x) = \tan^2(x) \times \sin^2(x)\) holds true for all values of \(x\) where the trigonometric functions are defined.

Would you like to explore this further or do you have any questions?

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Here are some related questions:
1. What are the domain restrictions for the equation \(\tan^2(x) - \sin^2(x) = \tan^2(x) \times \sin^2(x)\)?
2. How would the equation change if we replace \(\tan(x)\) with \(\cot(x)\)?
3. Can you derive similar identities using \(\cos(x)\) and \(\sin(x)\)?
4. What happens to the identity if we add \(\cos^2(x)\) to both sides of the original equation?
5. How does the identity behave at specific angles like \(x = \frac{\pi}{4}\) or \(x = \frac{\pi}{6}\)?

**Tip:** When simplifying trigonometric equations, substituting known identities like \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) often helps in simplifying the problem!

Explore the solution to the trigonometric equation tan^2(x) - sin^2(x) = tan^2(x) * sin^2(x) step-by-step. Understand the application of basic trigonometric identities and Pythagorean identities in solving this equation. Suitable for students in Grades 10-12.

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