Final answer:
The expression 1 - cos(45°) / sin(45°) simplifies to sin(45°) by using algebraic manipulation and the Pythagorean trigonometric identity.
Step-by-step explanation:
To simplify the expression using half-angle identities without evaluating, we can start by recognizing that the half-angle identities are a subset of trigonometric identities that relate the sine and cosine of half-angles to one another and to the original angles. However, looking at the given expression 1 - cos(45°) / sin(45°), we can simplify this directly without using half-angle identities just through algebraic manipulation and the use of basic trigonometric identities.
The expression can be rewritten using the Pythagorean identity sin²(θ) + cos²(θ) = 1, which we can rearrange to sin²(θ) = 1 - cos²(θ). For the angle 45°, we have sin(45°) = cos(45°), and thus the expression simplifies to:
sin(45°) = (1 - cos(45°)) / sin(45°)
= sin²(45°) / sin(45°)
= sin(45°)
Therefore, the simplified form of the expression is just sin(45°), which is a direct result of basic trigonometric identities rather than half-angle identities.