Which half-angle formula represents cos(x/2)?

The half-angle formula for cosine gives a way to find the cosine of half of an angle in terms of the trigonometric functions of the original angle. Specifically, the formula for cos(x/2) is derived from the identity for cosine in terms of sine.

The correct half-angle formula states that cos(x/2) can be expressed as the square root of the average of (1 + cos(x)) and (1 - sin(x)). When focusing specifically on the sine component, the formula simplifies to plus or minus the square root of (1 + sin(x))/2.

This representation is based on the fundamental identities of trigonometric functions and helps in simplifying problems involving angles. The presence of the plus or minus sign indicates that cos(x/2) can be either positive or negative, depending on the quadrant in which x/2 lies.

Thus, the formula plus or minus square root ((1 + sin(x))/2) accurately reflects the relationship between the half-angle cosine function and the sine of the original angle, confirming that this option is indeed the correct representation of cos(x/2).