https://file.aiquickdraw.com/mathbot/file/1726724997614ohf1xix9.jpg

It looks like the image contains a trigonometric equation and a proof.

The expression is as follows:

\[
\frac{7 - 6 \sin^2 \theta}{\sin^2 \theta} = 7 \csc^2 \theta - 6
\]

### Proof breakdown:
1. Start with the left-hand side of the equation: 
   \[
   \frac{7 - 6 \sin^2 \theta}{\sin^2 \theta}
   \]
   
2. Split the terms in the numerator:
   \[
   \frac{7}{\sin^2 \theta} - \frac{6 \sin^2 \theta}{\sin^2 \theta}
   \]

3. Simplify the second term:
   \[
   \frac{7}{\sin^2 \theta} - 6
   \]

4. Recognize that \(\frac{1}{\sin^2 \theta}\) is the definition of \(\csc^2 \theta\), so:
   \[
   7 \csc^2 \theta - 6
   \]

Thus, the left-hand side simplifies to the right-hand side of the equation, proving the identity.

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Would you like more details on any of the steps?

Here are 5 related questions to explore:

1. What is the relationship between \(\sin^2 \theta\) and \(\csc^2 \theta\)?
2. How would this proof differ if different trigonometric functions were used?
3. Can this identity be generalized to other forms of trigonometric identities?
4. What other identities involve \(\csc^2 \theta\)?
5. How does this proof relate to Pythagorean identities?

**Tip**: Always look for opportunities to break complex fractions into simpler parts when working with trigonometric identities.

Prove: \frac{7 - 6 \sin^2 \theta}{\sin^2 \theta} = 7 \csc^2 \theta - 6

Learn how to prove the trigonometric identity \frac{7 - 6 \sin^2 \theta}{\sin^2 \theta} = 7 \csc^2 \theta - 6 using basic trigonometric identities. This step-by-step solution is suitable for high school students in Grades 10-12.

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