Everyone knows the identity:
\[\sin^2\theta + \cos^2\theta = 1\]It implies two equivalent identities:
\[\begin{aligned} \tan^2\theta + 1 = \sec^2\theta \\ 1 + \cot^2\theta = \csc^2\theta \end{aligned}\]Rewriting and factoring gives
\[\begin{aligned} \sec^2\theta - \tan^2\theta = (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \\ \csc^2\theta - \cot^2\theta = (\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1 \end{aligned}\]So whenever defined,
It looks straightforward and perhaps useless at first glance. In fact, this reciprocal relationship allows us to multiply by a convenient form of 1 when evaluating $\int\sec(x)\,\mathrm(d)x$ and $\int\csc(x)\,\mathrm(d)x$. ref.
\[\begin{aligned} \int \sec(x) \,\mathrm{d}x = \ln(\vert \sec(x)+\tan(x) \vert) + C = - \ln(\vert \sec(x)-\tan(x) \vert) + C \\ \int \csc(x) \,\mathrm{d}x = -\ln(\vert \csc(x)+\cot(x) \vert) + C = \ln(\vert \csc(x)-\cot(x) \vert) + C \end{aligned}\]