Tan, Cot, Sec, and Csc Tan, Cot, Sec, and Csc
Introduction
We will prove the following:
\begin{align*}
\frac{d}{dx}\tan(x)&=\sec^2(x) \\\\
\frac{d}{dx}\cot(x)&=-\csc^2(x) \\\\
\frac{d}{dx}\sec(x)&=\tan(x)\sec(x) \\\\
\frac{d}{dx}\csc(x)&=-\cot(x)\csc(x)
\end{align*}
Preliminary Reminder
\sin^2(x)+\cos^2(x)=1
Proof of the Derivative of Tan(x)
\begin{align*}
\frac{d}{dx}\tan(x)&=\frac{d}{dx}\frac{\sin(x)}{\cos(x)} \\\\
&=\frac{\cos(x) \cdot \frac{d}{dx}[\sin(x)] - \sin(x) \cdot \frac{d}{dx}[\cos(x)]}{\cos^2(x)} \\\\
&=\frac{\cos(x) \cdot \cos(x) - \sin(x) \cdot [-\sin(x)]}{\cos^2(x)} \\\\
&=\frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} \\\\
&=\frac{1}{\cos^2(x)} \\\\
&=\sec^2(x)
\end{align*}
Proof of the Derivative of Cot(x)
\begin{align*}
\frac{d}{dx}\cot(x)&=\frac{d}{dx}\frac{\cos(x)}{\sin(x)} \\\\
&=\frac{\sin(x) \cdot \frac{d}{dx}[\cos(x)] - \cos(x) \cdot \frac{d}{dx}[\sin(x)]}{\sin^2(x)} \\\\
&=\frac{\sin(x) \cdot [-\sin(x)] - \cos(x) \cdot \cos(x)}{\sin^2(x)} \\\\
&=\frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)} \\\\
&=\frac{-(\sin^2(x) + \cos^2(x))}{\sin^2(x)} \\\\
&=\frac{-1}{\sin^2(x)} \\\\
&=-\csc^2(x)
\end{align*}
Proof of the Derivative of Sec(x)
The next two proofs are going to use the chain rule for one step. We will prove the chain rule in the next lesson, so it seems fine to use it here instead of waiting.
\begin{align*}
\frac{d}{dx}\sec(x)&=\frac{d}{dx}\frac{1}{\cos(x)} \\\\
&=\frac{d}{dx}\cos(x)^{-1} \\\\
&= {-1}\cos(x)^{-2} \cdot \frac{d}{dx}\cos(x) \text{ (Chain Rule)} \\\\
&= {-1}\cos(x)^{-2} \cdot [-\sin(x)] \\\\
&=\frac{\sin(x)}{\cos^2(x)} \\\\
&=\frac{\sin(x)}{\cos(x)} \cdot \frac{1}{\cos(x)} \\\\
&=\tan(x)\sec(x)
\end{align*}
Proof of the Derivative of Csc(x)
\begin{align*}
\frac{d}{dx}\csc(x)&=\frac{d}{dx}\frac{1}{\sin(x)} \\\\
&=\frac{d}{dx}\sin(x)^{-1} \\\\
&= {-1}\sin(x)^{-2} \cdot \frac{d}{dx}\sin(x) \text{ (Chain Rule)} \\\\
&= {-1}\sin(x)^{-2} \cdot \cos(x) \\\\
&=\frac{-\cos(x)}{\sin^2(x)} \\\\
&=\frac{-\cos(x)}{\sin(x)} \cdot \frac{1}{\sin(x)} \\\\
&=-\cot(x)\csc(x)
\end{align*}
These derivative definitions are used often and are included in the
Appendix - Common Derivatives.