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The problem in the image asks to prove the following identity:

\[
1 + \tan^2 \theta = \frac{\tan^2 \theta}{\sin^2 \theta}
\]

### Proof:

1. Start with the left-hand side (LHS) of the equation:

\[
1 + \tan^2 \theta
\]

Using the identity \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \), substitute for \( \tan^2 \theta \):

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

2. Combine into a single fraction:

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

3. Use the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

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

Now, this is equal to \( \sec^2 \theta \), but let's hold that thought for now and move to the right-hand side (RHS).

4. The right-hand side (RHS) is:

\[
\frac{\tan^2 \theta}{\sin^2 \theta}
\]

Substitute \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \):

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

5. Simplify the fraction:

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

6. Now both sides are equal:

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

Thus, the identity is proven.

### Conclusion:
\[
1 + \tan^2 \theta = \frac{\tan^2 \theta}{\sin^2 \theta}
\]

---

Would you like a further breakdown of any step, or do you have any other questions?

### Related questions:
1. How do we derive the identity \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)?
2. What is the significance of the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \)?
3. How do we use the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) in trigonometry?
4. Can this identity be applied in solving trigonometric equations?
5. How can this identity be visualized on the unit circle?

#### Tip:
When proving trigonometric identities, always keep the fundamental identities (like Pythagorean, quotient, and reciprocal identities) in mind. They are very useful!

Prove that 1 + tan^2(θ) = tan^2(θ) / sin^2(θ)

Learn how to prove the trigonometric identity 1 + tan^2(θ) = tan^2(θ) / sin^2(θ) with a step-by-step solution using key trigonometric identities such as the Pythagorean identity and the quotient identity. Ideal for students in Grades 10-12.

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