Perform the addition or subtraction.

tanx - sec^2x/tanx

tan^2(x)/tan(x) - sec^2x/tanx =

tan^2x sec^2x / tanx

then I use the identity 1+tan^2u=sec^2u

I do not know what to do at this point.

from you second line

tan^2(x)/tan(x) - sec^2x/tanx
= (tan^2 x - sec^2 x)/tanx
= (tan^2 x - (1 + tan^2 x))/tanx
= -1/tanx or -cotx or -sinx/cosx

To simplify the expression further, you can use the trigonometric identity 1 + tan^2(u) = sec^2(u). Let's apply this identity to the expression you have:

tan^2(x) sec^2(x) / tan(x)

Using the identity 1 + tan^2(u) = sec^2(u), we can rewrite tan^2(x) as sec^2(x) - 1:

(sec^2(x) - 1) sec^2(x) / tan(x)

Next, let's multiply sec^2(x) by (sec^2(x) - 1):

sec^4(x) - sec^2(x) / tan(x)

Now, we have a difference of squares in the numerator, so we can factor it:

(sec^2(x) - sec(x))(sec^2(x) + sec(x)) / tan(x)

Finally, we can simplify further by factoring out a sec(x) from both terms in the numerator:

sec(x)(sec(x) - 1)(sec(x) + 1) / tan(x)

So, the simplified expression is sec(x)(sec(x) - 1)(sec(x) + 1) / tan(x).

Remember, when simplifying expressions involving trigonometric functions, it's helpful to use trigonometric identities and algebraic manipulations.