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