Since tan ( x ) = sin ( x ) cos ( x ) {\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}} , we can find its derivative by the usual rule for differentiating a fraction:
Similarly,
![{\displaystyle {\frac {d}{dx}}\left[{\frac {\sin(x)}{\cos(x)}}\right]={\frac {\cos(x)\cdot \cos(x)+\sin(x)\cdot \sin(x)}{\cos ^{2}(x)}}={\frac {1}{\cos ^{2}(x)}}=\sec ^{2}(x)={1+\tan ^{2}(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f80daf15c5a521b668f9572b393dd30e686677f8)
![{\displaystyle {\frac {d}{dx}}{\bigl [}\cot(x){\bigr ]}=\csc ^{2}(x)=1+\cot ^{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9764368d5895c6dd7fb04438c216bd40f64da5)
![{\displaystyle {\frac {d}{dx}}{\bigl [}{\text{sec}}(x){\bigr ]}={\frac {\sin(x)}{\cos ^{2}(x)}}=\tan(x)\sec(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/290926eb3eddec3ccddcf9350ce6cd55d1369309)
![{\displaystyle {\frac {d}{dx}}{\bigl [}\csc(x){\bigr ]}=-{\frac {\cos(x)}{\sin ^{2}(x)}}=-\cot(x)\csc(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e93725c4f5209131b042e536118c91d4d24002a)