Final answer:
The trigonometric expression sin^2θ - secθcosθ + cos^2θ simplifies to 1 using fundamental trigonometric identities, such as the Pythagorean identity and the reciprocal of cosine.
Step-by-step explanation:
The student's question involves simplifying a trigonometric expression: sin2θ - secθcosθ + cos2θ. To simplify this, we utilize fundamental trigonometric identities. Firstly, recall that secθ = 1/cosθ, which means the term secθcosθ simplifies to 1. Secondly, use the Pythagorean identity sin2θ + cos2θ = 1 to replace the sum of squares. With these substitutions, the expression simplifies as follows:
Substitute secθ with 1/cosθ and simplify secθcosθ to 1,
Apply the Pythagorean identity sin2θ + cos2θ = 1,
The final simplified expression is simply 1.
Therefore, the original expression simplifies to 1.