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To find the correct expression for \( \sin x \) in terms of \( \cos x \), let's analyze each option using trigonometric identities. 

One helpful identity to recall is the Pythagorean identity:
\[
\sin^2 x + \cos^2 x = 1.
\]
From this, we can express \( \sin^2 x \) as:
\[
\sin^2 x = 1 - \cos^2 x.
\]
Taking the square root of both sides gives:
\[
\sin x = \pm \sqrt{1 - \cos^2 x}.
\]

Now, let’s evaluate each option:

1. **Option A: \( \sin x = \frac{1}{1 - \cos x} \)**
   - This does not match our derived expression for \( \sin x \) and is not equivalent to any standard trigonometric identity. This option is incorrect.

2. **Option B: \( \sin x = \pm \sqrt{1 - \cos^2 x} \)**
   - This matches exactly with our derived expression from the Pythagorean identity. This option is correct.

3. **Option C: \( \sin x = (1 - \cos x)(1 + \cos x) \)**
   - Expanding \( (1 - \cos x)(1 + \cos x) \) gives \( 1 - \cos^2 x \), which equals \( \sin^2 x \), not \( \sin x \). This is incorrect.

4. **Option D: \( \sin x = \pm \sqrt{1 - \cos x} \)**
   - This does not match our expression because it involves \( \sqrt{1 - \cos x} \), not \( \sqrt{1 - \cos^2 x} \). This option is incorrect.

**Conclusion:** The correct answer is **Option B: \( \sin x = \pm \sqrt{1 - \cos^2 x} \)**.

Would you like more details or have any questions?

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### Related Questions:
1. How do we derive other trigonometric identities involving both \( \sin \) and \( \cos \)?
2. What is the significance of the \(\pm\) sign in trigonometric expressions?
3. How can we express \( \cos x \) in terms of \( \sin x \)?
4. How would this identity change if we were given a different trigonometric function, like \( \tan x \)?
5. Can we use a similar approach to express \( \sin 2x \) in terms of \( \cos 2x \)?

### Tip:
Remember that the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) is one of the most fundamental identities in trigonometry and can be very useful in rewriting expressions.

Use trigonometric identities to write sin x in terms of cos x. Choose the correct expression equal to sin x from the options.

Learn how to express sin x in terms of cos x using trigonometric identities. This solution covers the use of the Pythagorean identity to derive sin x in terms of cos x. Suitable for high school students studying trigonometry.

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