To simplify the expression, we can first combine the fractions:

(1 - sin α)/(cos α) - (cos α)/(1 + sin α)

Next, we can find a common denominator by multiplying the right fraction by (cos α)/(cos α):

[(1 - sin α)/(cos α)] - [(cos α)/(1 + sin α)] x [(cos α)/(cos α)]
= (1 - sin α)/(cos α) - (cos^2 α)/(cos α(1 + sin α))

Now, we can simplify further by factoring out a cos α in the second term:

= (1 - sin α)/(cos α) - (cos α)/(1 + sin α)

This is the simplified expression.