You can use a
![](https://latex.codecogs.com/png.latex?\bg_white u=\sin\theta)
substitution after the simplification but it's not that necessary IMHO.
After the
![](https://latex.codecogs.com/png.latex?\bg_white \sec\theta)
sub, you end up with
![](https://latex.codecogs.com/png.latex?\bg_white \int \cot\theta \csc\theta d\theta)
. If we see patterns in the integrals of trigonometric functions, this is easy to integrate.
![](https://latex.codecogs.com/png.latex?\bg_white \int \sec\theta \tan\theta d\theta=\sec\theta)
. It can be deduced that
![](https://latex.codecogs.com/png.latex?\bg_white \int \cot\theta \csc\theta d\theta=-\csc\theta)
. A similar relationship applies for
![](https://latex.codecogs.com/png.latex?\bg_white \int \sec^2x\; dx =\;\tan{x})
and
![](https://latex.codecogs.com/png.latex?\bg_white \int \csc^2x\;dx= - \;\cot{x} dx)
. These relationships are very useful when determining integrals of trig functions. It's very nice how you can 'co' all the functions and then put a minus sign out the front - if you know what I mean.
Additionally,
Looks similar to the trig identities?
+Cs, obviously.