Trigonometry/Derivative of Tangent

< Trigonometry

Since \tan(x)=\frac{\sin(x)}{\cos(x)} , we can find its derivative by the usual rule for differentiating a fraction:

\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)} .

Similarly,

\frac{d}{dx}\bigl[\cot(x)\bigr]=\csc^2(x)=1+\cot^2(x)
\frac{d}{dx}\bigl[\text{sec}(x)\bigr]=\frac{\sin(x)}{\cos^2(x)}=\tan(x)\sec(x)
\frac{d}{dx}\bigl[\csc(x)\bigr]=-\frac{\cos(x)}{\sin^2(x)}=-\cot(x)\csc(x)