https://file.aiquickdraw.com/mathbot/file/1725471320284qe5ptx5a.jpg

The expression from the image is:

\[
\frac{\sin A}{1 - \cos A} + \frac{1 - \cos A}{\sin A} = 2 \csc A
\]

Let's solve this step-by-step.

### Step-by-Step Solution

1. **Combine the fractions on the left-hand side:**

\[
\frac{\sin A}{1 - \cos A} + \frac{1 - \cos A}{\sin A} = \frac{\sin^2 A + (1 - \cos A)^2}{\sin A (1 - \cos A)}
\]

2. **Expand the numerator:**

\[
\sin^2 A + (1 - \cos A)^2 = \sin^2 A + (1 - 2 \cos A + \cos^2 A)
\]

Combine like terms:

\[
= \sin^2 A + \cos^2 A - 2 \cos A + 1
\]

3. **Use the Pythagorean identity:**

\[
\sin^2 A + \cos^2 A = 1
\]

So, the numerator becomes:

\[
1 - 2 \cos A + 1 = 2(1 - \cos A)
\]

4. **Substitute the simplified numerator back:**

\[
\frac{2(1 - \cos A)}{\sin A (1 - \cos A)}
\]

Cancel \(1 - \cos A\) from the numerator and denominator:

\[
= \frac{2}{\sin A}
\]

5. **Use the cosecant identity:**

\[
\frac{2}{\sin A} = 2 \csc A
\]

### Conclusion

The left-hand side simplifies to the right-hand side:

\[
\frac{\sin A}{1 - \cos A} + \frac{1 - \cos A}{\sin A} = 2 \csc A
\]

### The given equation is verified to be true.

Would you like more details, or have any questions about this solution?

Here are 5 related questions to explore further:
1. What is the Pythagorean identity in trigonometry?
2. How do you prove trigonometric identities?
3. What is the relationship between sine, cosine, and cosecant?
4. Can you solve another trigonometric identity involving secant or tangent?
5. How do you convert between different trigonometric functions?

**Tip:** When simplifying trigonometric expressions, always look for opportunities to use fundamental identities like the Pythagorean identity or reciprocal identities.

\( \frac{\sin A}{1 - \cos A} + \frac{1 - \cos A}{\sin A} = 2 \csc A \)

Learn how to verify the trigonometric identity \( \frac{\sin A}{1 - \cos A} + \frac{1 - \cos A}{\sin A} = 2 \csc A \) using step-by-step simplifications. This problem involves key trigonometric concepts such as the Pythagorean identity and the cosecant identity. Suitable for students in Grades 9-12.

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