Final answer:
The simplified expression of (secα+tanα−1)(secα−tanα+1) equals 1, after utilizing trigonometric identities and simplifying. None of the provided options match this result.
Step-by-step explanation:
The given expression, (secα+tanα−1)(secα−tanα+1), can be simplified by recognizing it as the product of a sum and a difference of two terms. This pattern corresponds to the algebraic identity (a+b)(a-b) which equals to a2 - b2. Substituting secα and tanα into the identity, we obtain secα2 − tanα2. Using the trigonometric identities, we know that secα = 1/cosα and tanα = sinα/cosα, which leads us to (1/cosα)2 − (sinα/cosα)2. Simplifying further, this becomes 1/cos2α − sin2α/cos2α which simplifies to (1-sin2α)/cos2α. From the Pythagorean identity, we have that 1 − sin2α is cos2α, so the expression simplifies to cos2α/cos2α which is simply 1. Therefore, the simplified expression equals 1, which is not listed among the provided options A) 2sinα B) 2cosα C) 2tanα D) 2cotα, indicating a possible typo or error in the question.