Trigonometry/Derivative of Tangent

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,

$\frac{d}{dx}[\cot (x)]={\csc }^2(x)=1+{\cot }^2(x)$

$\frac{d}{dx}[\text{sec}(x)]=\frac{\sin (x)}{{\cos }^2(x)}=\tan (x)\sec (x)$

$\frac{d}{dx}[\csc (x)]=-\frac{\cos (x)}{{\sin }^2(x)}=-\cot (x)\csc (x)$