Solution:

Simplifying the given expression:

(cosecx-sinx)2 = (1/sinx - sinx)2
= ((1-sin2x)/sinx)2
= (cos2x/sinx)2
= cos2x/sin2x

(secx-cosx)2 = (1/cosx - cosx)2
= ((1-cos2x)/cosx)2
= (sin2x/cosx)2
= sin2x/cos2x

(cotx-tanx)2 = (cosx/sinx - sinx/cosx)2
= ((cos2x - sin2x)/sinxcosx)2
= ((cosx+sinx)(cosx-sinx)/sinxcosx)2
= ((cosx+sinx)/sinx)2 ((cosx-sinx)/cosx)2
= (1+tanx/cotx)2 (1-tanx/cotx)2
= (cotx+tanx)2 (cotx-tanx)2

Therefore, the given expression becomes:

cos2x/sin2x - sin2x/cos2x - (cotx+tanx)2 (cotx-tanx)2

Integration of sinx/cos2x:

We can solve the integration of sinx/cos2x using the substitution method. Let u = cosx, then du/dx = -sinx and dx = -du/sin(x). Therefore, the integral becomes:

∫sinx/cos2x dx = ∫-du/u2
= u-1 + C
= cosx-1 + C

Hence, the solution to the integration of sinx/cos2x is cosx-1 + C.

Summary:

- To simplify the given expression, we use trigonometric identities to obtain cos2x/sin2x - sin2x/cos2x - (cotx+tanx)2 (cotx-tanx)2.
- The integration of sinx/cos2x can be solved using the substitution method. We let u = cosx, then du/dx = -sinx and dx = -du/sin(x). We obtain the solution cosx-1 + C.