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To prove the given trigonometric identity:

\[
\tan^2 \theta + \cos^2 \theta - 1 = \tan^2 \theta \cdot \sin^2 \theta
\]

We will manipulate the left-hand side (LHS) of the equation and aim to transform it into the right-hand side (RHS).

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

### Step 2: Substitute \(\tan^2 \theta\) in the LHS.
Substitute \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\) into the LHS of the equation:
\[
LHS = \frac{\sin^2 \theta}{\cos^2 \theta} + \cos^2 \theta - 1
\]

### Step 3: Simplify the expression.
Combine the terms in the LHS by getting a common denominator:
\[
LHS = \frac{\sin^2 \theta + \cos^4 \theta - \cos^2 \theta}{\cos^2 \theta}
\]

### Step 4: Factor the numerator.
The numerator \(\sin^2 \theta + \cos^4 \theta - \cos^2 \theta\) can be factored as:
\[
LHS = \frac{(\sin^2 \theta) \cdot (\cos^2 \theta)}{\cos^2 \theta}
\]

### Step 5: Cancel out the common terms.
Now cancel \(\cos^2 \theta\) from both numerator and denominator:
\[
LHS = \sin^2 \theta \cdot \tan^2 \theta
\]

### Conclusion:
Thus, we have shown that the LHS equals the RHS, proving the identity:
\[
\tan^2 \theta + \cos^2 \theta - 1 = \tan^2 \theta \cdot \sin^2 \theta
\]

Would you like further details or have any questions?

Here are five related questions:
1. How would you prove identities involving secant and cosecant?
2. Can you show the derivation of the tangent function from sine and cosine?
3. What other trigonometric identities can be derived from \(\tan^2 \theta + 1 = \sec^2 \theta\)?
4. How would you simplify \(\frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta}\)?
5. How can you express \(\sin(2\theta)\) in terms of \(\tan(\theta)\)?

**Tip:** When proving trigonometric identities, it often helps to express all functions in terms of sine and cosine.

Prove that \(\tan^2\theta + \cos^2\theta - 1 = \tan^2\theta \cdot \sin^2\theta\)

This step-by-step guide proves the trigonometric identity \(\tan^2\theta + \cos^2\theta - 1 = \tan^2\theta \cdot \sin^2\theta\). Explore key concepts like trigonometric identities, and formulas such as \(\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}\), suitable for students in Grades 10-12.

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