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To prove the identity:

\[
\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta
\]

we'll break it down step-by-step:

### Step 1: Express in terms of sine and cosine
Let's rewrite \(\csc^2 \theta\) and \(\tan^2 \theta\) using sine and cosine:
- \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\)
- \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\)

Substituting these into the left side of the equation gives:

\[
\frac{1}{\sin^2 \theta} \cdot \frac{\sin^2 \theta}{\cos^2 \theta} - 1
\]

### Step 2: Simplify the expression
The \(\sin^2 \theta\) terms cancel out in the first term:

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

### Step 3: Rewrite in terms of \(\sec^2 \theta\)
Since \(\frac{1}{\cos^2 \theta} = \sec^2 \theta\), we now have:

\[
\sec^2 \theta - 1
\]

### Step 4: Use a Pythagorean identity
From the Pythagorean identity, we know:

\[
\sec^2 \theta - 1 = \tan^2 \theta
\]

Thus, the left side simplifies to \(\tan^2 \theta\), which matches the right side of the identity.

### Conclusion
The identity is proven as:

\[
\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta
\]

Would you like more details on this proof or have any other questions?

Here are five related questions that build on this topic:
1. What are the Pythagorean identities, and how can they be applied to similar problems?
2. How can we use trigonometric identities to simplify complex expressions?
3. Why is it useful to convert trigonometric functions to sines and cosines when proving identities?
4. What are some common strategies for verifying trigonometric identities?
5. How do reciprocal identities play a role in simplifying trigonometric expressions?

**Tip:** When proving trigonometric identities, try converting everything to sines and cosines first—this often makes the algebra easier to work with.

Prove the identity: csc^2(θ) tan^2(θ) - 1 = tan^2(θ)

Prove the trigonometric identity csc^2(θ) tan^2(θ) - 1 = tan^2(θ) using fundamental identities and algebraic manipulation. This step-by-step guide covers essential trigonometric concepts and is ideal for students in Grades 10-12.

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