Verify (tanx-cotx)/sinxcosx = sec^2x-csc^2x

Tan= sin/cos
cot= cos/sin
sec= (1/cos)^2
csc= (1/sin)^2

((sinx/cosx)-(cosx/sinx))/cosxsinx
now multiply cosxsinx into the denominator so you get
(sinx/cos^2xsinx)-(cosx/sin^2xcosx)
the numerator and the denominator cancels so you have 
(1/cos^2x)-(1/sin^2x) which is the same as  sec^2x-csc^2x