reciprocal trigonometric function

sec for example

${\displaystyle \sec x=\frac{1}{\cos x}}$

${\displaystyle D_x\sec x=\sec x\tan x}$

${\displaystyle \int \sec xdx=\tan x+c}$

${\displaystyle \csc x=\frac{1}{\sin x}}$${\displaystyle \cot x=\frac{1}{\tan x}}$

${\displaystyle D_x\csc x=-\csc x\cot x}$${\displaystyle D_x\cot x=-{\csc }^{2}x}$

${\displaystyle \int \csc xdx=-\cot x+c}$${\displaystyle \int \cot xdx=\ln |\sin x|+c}$

proof for the integration of ${\displaystyle \cot x}$

${\displaystyle \cot x=\frac{\cos x}{\sin x}}$,

${\displaystyle \int \cot xdx=\int \frac{\cos x}{\sin x}dx}$,

let ${\displaystyle u=\sin x}$, then ${\displaystyle du=\cos xdx}$

${\displaystyle \int \cot xdx=\int \frac{1}{u}du=\ln |u|+c=\ln |\sin x|+c}$

${\displaystyle \int {\sec }^{3}xdx}$

${\displaystyle \int \tan xdx=}$