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?

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.