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