Answer:
To continue we can expand the given expression using the identities:
secA = 1/cosA
tanA = sinA/cosA
We have:
(secA + tanA - 1) * (secA - tanA + 1)
= (1/cosA + sinA/cosA - 1) * (1/cosA - sinA/cosA + 1)
Now let's simplify the expression. Multiplying the numerators and denominators in each term:
= (1 + sinA - cosA) * (1 - sinA + cosA) / cosA * cosA
Multiplying the binomials:
= (1 - sinA + cosA + sinA - sin^2A + sinAcosA - cosA + sinAcosA + cos^2A) / cos^2A
Simplifying:
= (2 - sin^2A + 2sinAcosA + cos^2A) / cos^2A
Now we can use the Pythagorean identities sin^2A + cos^2A = 1 and 2sinAcosA = sin(2A):
= (2 - 1 + sin(2A)) / cos^2A
= (1 + sin(2A)) / cos^2A
Therefore the simplified expression is (1 + sin(2A)) / cos^2A.