Answer: We can simplify the expression using trigonometric identities:
tan a + sec a - 1 - (tan a - sec a + 1) / (1 + sin a cos a)
= tan a + sec a - 1 - (tan a - sec a + 1) / cos^2 a
= (sin a / cos a + 1 / cos a - cos a / cos a) - [(sin a / cos a - 1 / cos a - cos a / cos a) / cos^2 a]
= (sin a + 1 - cos a) / cos a - [(sin a - 1 - cos^2 a) / cos^3 a]
= [(sin a + 1 - cos a) cos^2 a - (sin a - 1 - cos^2 a)] / cos^4 a
= [sin a cos^2 a + cos^2 a - cos a cos^2 a - sin a + 1 + cos^2 a] / cos^4 a
= (2 cos^2 a - sin a + 1) / cos^4 a
Now, we can simplify the expression further using the identity:
1 + tan^2 a = sec^2 a
tan^2 a = sec^2 a - 1
tan a + sec a - 1 = (tan^2 a + 1) / (sec a + tan a - 1)
= (sec^2 a - 1 + 1) / (sec a + tan a - 1)
= sec a / (sec a + tan a - 1)
tan a - sec a + 1 = -(sec a - tan a - 1)
= -(1 / (sec a - tan a + 1))
Substituting these values in the original expression, we get:
(sec a / (sec a + tan a - 1)) - (-1 / (sec a - tan a + 1)) / (1 + sin a cos a)
= (sec a (sec a - tan a + 1) + (tan a - 1)) / ((sec a + tan a - 1)(sec a - tan a + 1)(1 + sin a cos a))
= [(sec^2 a - sin a + 1) + (tan a - 1)] / (cos^2 a (1 + sin a cos a))
= [(2 cos^2 a - sin a + 1)] / (cos^4 a (1 + sin a cos a))
Thus, we have simplified the given expression to (2 cos^2 a - sin a + 1) / (cos^4 a (1 + sin a cos a)).
Explanation: