What is the formula for cos(x/2)?

The formula for cos(x/2) is derived from the half-angle identities in trigonometry. Specifically, the half-angle formula states that:

[

\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}}.

]

This identity is crucial because it relates the cosine of an angle to the cosine of half that angle. When using this formula, the plus or minus sign indicates that the cosine function can be positive or negative depending on the quadrant in which the angle x/2 lies.

For example, if x is in the range where x/2 is in the first two quadrants (0 to π), then cos(x/2) will be positive, while if x is in the range of the third or fourth quadrants (π to 2π), cos(x/2) will be negative. This is important for determining the correct sign based on the values of x.

The other options do not accurately represent the half-angle formula or misapply other trigonometric identities. As a result, B is the accurate representation of the relationship for cos(x/2).