To prove the trigonometric identity, starting with the left side is recommended, transforming it using algebra and standard identities. The discussion emphasizes converting everything to sines and cosines, particularly using the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1. By rearranging this identity, one can express 1 - sin²(θ) as cos²(θ), aiding in simplification. The final steps involve recognizing that sec(θ) is equal to 1/cos(θ), leading to the conclusion that sec(θ)cos²(θ) simplifies to cos(θ). Understanding these transformations is key to proving the identity successfully.