For the formula tan(x/2), which expression applies?

The expression that correctly represents tan(x/2) is indeed sin(x)/(1 + cos(x)). This relationship is derived from the half-angle identities in trigonometry.

To understand why this is the correct expression, consider the half-angle identity for tangent. The half-angle identity states that:

[

\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)}

]

This identity emerges from the sine and cosine definitions of tangent and can be derived using the double-angle formulas for sine and cosine. Specifically, if we let ( u = x/2 ), then ( x = 2u ) and we have:

[

\tan(u) = \frac{\sin(2u)}{\cos(2u)} = \frac{2\sin(u)\cos(u)}{1 - 2\sin^2(u)}

]

By manipulating the double-angle identities and properties of sine and cosine, one can establish that indeed,:

[

\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)}

]

Each other option does not provide the correct relationship for tan(x/2).