In terms of x, how is sec(x/2) expressed?

The expression for sec(x/2) can be understood by recalling the definition of the secant function and using trigonometric identities. The secant function, sec(θ), is defined as the reciprocal of the cosine function, so sec(x/2) is equal to 1/cos(x/2).

To express sec(x/2) in terms of cos(x), one can utilize the half-angle identity for cosine, which states that:

[

\cos(x/2) = \sqrt{\frac{1 + \cos(x)}{2}}

]

From this identity, sec(x/2) can be expressed as:

[

sec(x/2) = \frac{1}{\cos(x/2)} = \frac{1}{\sqrt{\frac{1 + \cos(x)}{2}}}

]

To simplify this, we multiply the numerator and the denominator by the square root to rationalize:

[

sec(x/2) = \frac{\sqrt{2}}{\sqrt{1 + \cos(x)}}

]

Squaring sec(x/2) gives us an expression involving the square root of both the numerator and the denominator, leading to another type of expression that simplifies to involve