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Which of the following is not an identity for [tex]\tan \left(\frac{x}{2}\right)[/tex]?

A. [tex]\frac{1-\cos x}{\sin x}[/tex]

B. [tex]\frac{\sin x}{1+\cos x}[/tex]

C. [tex]\frac{\cos x}{1-\sin x}[/tex]

D. [tex]\pm \sqrt{\frac{1-\cos x}{1+\cos x}}[/tex]


Sagot :

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].