To determine which of the given expressions is not an identity for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex], let's analyze each one in detail.
The correct identities for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex] are well-known and can be derived from trigonometric identities and half-angle formulas. They include:
1. [tex]\(\tan \left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x}\)[/tex]
2. [tex]\(\tan \left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}\)[/tex]
3. [tex]\(\tan \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\)[/tex]
Given these identities, we can verify each option:
A. [tex]\(\frac{1 - \cos x}{\sin x}\)[/tex]
This is a known identity for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex].
B. [tex]\(\frac{\sin x}{1 + \cos x}\)[/tex]
This is another known identity for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex].
C. [tex]\(\frac{\cos x}{1 - \sin x}\)[/tex]
This expression does not match any known identity for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex]. Therefore, it's the incorrect one.
D. [tex]\(\pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\)[/tex]
This is also a known identity for [tex]\(\tan \left(\frac{x}{2}\right)\)[/tex].
Therefore, the incorrect identity among the given options is:
C. [tex]\(\frac{\cos x}{1 - \sin x}\)[/tex]
So the answer is [tex]\(\boxed{3}\)[/tex].
