We are given the trigonometric equation:

\[
\csc^2x \cdot \tan^2x - 1 = \tan^2x
\]

Let's solve this step by step.

### Step 1: Recall the identity for \(\csc^2x\)
We know that:

\[
\csc^2x = 1 + \cot^2x
\]

This identity will help simplify the expression. Substituting \(\csc^2x\) in the given equation:

\[
(1 + \cot^2x) \cdot \tan^2x - 1 = \tan^2x
\]

### Step 2: Expand the left-hand side
Expand the left-hand side of the equation:

\[
\tan^2x + \cot^2x \cdot \tan^2x - 1 = \tan^2x
\]

### Step 3: Simplify \(\cot^2x \cdot \tan^2x\)
Since \(\cot x = \frac{1}{\tan x}\), we have:

\[
\cot^2x \cdot \tan^2x = 1
\]

So the equation becomes:

\[
\tan^2x + 1 - 1 = \tan^2x
\]

### Step 4: Simplify the equation
The terms \(+1\) and \(-1\) cancel each other out:

\[
\tan^2x = \tan^2x
\]

This is an identity, which means the equation is true for all values of \(x\) where the trigonometric functions are defined. Therefore, the equation holds for all values of \(x\) except those that would make \(\tan x\) or \(\csc x\) undefined.

### Final Answer:
The equation \(\csc^2x \cdot \tan^2x - 1 = \tan^2x\) is an identity and holds for all \(x\) where \(\tan x\) and \(\csc x\) are defined.

---

Do you have any questions or would you like further details on any step? Here are some related questions you might consider:

1. What values of \(x\) make \(\tan x\) and \(\csc x\) undefined?
2. Can you prove other trigonometric identities involving \(\tan x\) and \(\csc x\)?
3. How do the Pythagorean identities relate to this equation?
4. What are the implications of this being an identity rather than a conditional equation?
5. How does this identity change when considering radians vs. degrees?

**Tip**: Always check the domain of trigonometric functions to ensure solutions are valid within the range.

This page explains how to solve the trigonometric equation csc^2x tan^2x - 1 = tan^2x step by step. Using key identities like csc^2x = 1 + cot^2x, this solution is designed for students in Grades 10-12. Explore how trigonometric identities simplify the equation and understand the solution in detail.

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