Final answer:
The half-angle identities for sin²(x) and cos²(x) are used in integrating these functions and are derived from double-angle formulas. The identities express these functions' integrals in terms of the cosine function for easy evaluation.
Step-by-step explanation:
The half-angle identities used to integrate sin²(x) and cos²(x) are derived from the double-angle formulas for sine and cosine. For sine, the half-angle identity is sin²(x) = (1 - cos(2x))/2, and for cosine, it is cos²(x) = (1 + cos(2x))/2. These identities allow the integrals of sin²(x) and cos²(x) to be expressed in terms of the integral of a cosine function, which is straightforward to evaluate.
Using these identities, we can handle the periodic nature of these functions, as both the sine and cosine function's squares average out over a complete cycle, thus the integral over a period for cos²(x) is the same as for sin²(x).
Key identities mentioned in related material include the double-angle formulas:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)