Concept preview

What you will be able to do

1. 01State the derivatives of $\tan x$, $\cot x$, $\sec x$, and $\csc x$, and reproduce the co-function minus-sign pattern correctly.
2. 02Derive $\frac{d}{dx}\tan x=\sec^2 x$ and $\frac{d}{dx}\sec x=\sec x\tan x$ from $\tan x=\frac{\sin x}{\cos x}$ and $\sec x=\frac{1}{\cos x}$ using the quotient rule.
3. 03Evaluate these derivatives at the standard angles $0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3}$ to produce clean numerical values.

The sine and cosine derivatives are the two foundations; the other four trig derivatives are consequences. Because $\tan x=\frac{\sin x}{\cos x}$, $\cot x=\frac{\cos x}{\sin x}$, $\sec x=\frac{1}{\cos x}$, and $\csc x=\frac{1}{\sin x}$ are all quotients of sine and cosine, the quotient rule from 2.9 delivers every one of their derivatives. This lesson derives them so you understand where they come from, then organizes them so you can recall them reliably and evaluate them at the standard exam angles.