To simplify the given expression:
sin^2(p/2 + a) - cos^2(p/2 - a) / sin a + cos a

We will use the identities: sin^2(x) - cos^2(x) = -cos(2x) and sin(a + b) = sin a cos b + cos a sin b.

Substitute p/2 + a as x and p/2 - a as y.

sin^2(p/2 + a) - cos^2(p/2 - a) = -cos(2(p/2 + a))
= -cos(p + 2a)

sin a + cos a = sin (p/2)
= 1

Therefore, the simplified expression becomes:
-cos(p + 2a)/1
= -cos(p + 2a)