Derivatives of Reciprocal Trigonometric Functions

# Derivatives of Reciprocal Trigonometric Functions

We are going to look at even more derivative rules - this time for the reciprocal trigonometric functions.

 Theorem 1: The following functions have the following derivatives: a) If $f(x) = \sec x$, then $\frac{d}{dx} \sec x = \sec x \tan x$. b) If $f(x) = \csc x$, then $\frac{d}{dx} \csc x = -\csc x \cot x$. c) If $f(x) = cot x$, then $\frac{d}{dx} \cot x = -csc ^2 x$.
• Proof of a): Let $f(x) = \sec x$. Recall that the quotient rule is $\frac{f}{g} = \frac{gf' - fg'}{g^2}$ and that $\sec x = \frac{1}{\cos x}$. Applying the quotient rule, we get:
(1)
\begin{align} \frac{d}{dx} \sec x = \frac{\cos x \cdot 0 - 1 \cdot (-\sin x)}{\cos ^2 x} \\ \frac{d}{dx} \sec x = \frac{\sin x}{\cos ^2 x } \\ \frac{d}{dx} \sec x = \sec x \tan x \quad \blacksquare \end{align}
• Proof of b): Let $f(x) = \csc x$. Note that $\csc x = \frac{1}{\sin x}$. Applying the quotient rule we get:
(2)
\begin{align} \frac{d}{dx} \csc x = \frac{\sin x \cdot 0 - 1 \cdot \cos x}{\sin ^2 x } \\ \frac{d}{dx} \csc x = \frac{-\cos x}{\sin ^2 x} \\ \frac{d}{dx} \csc x = -\csc x \cot x \quad \blacksquare \end{align}
• Proof of c): Let $f(x) = \cot x$. Note that $\cot x = \frac{\cos x}{\sin x}$. Applying the quotient rule and using the trigonometric identity that $\sin ^2 x + \cos ^2 x = 1$, we get:
(3)
\begin{align} \frac{d}{dx} \cot x = \frac{\sin x \cdot (-\sin x) - \cos x \cdot \cos x}{\sin ^2 x} \\ \frac{d}{dx} \cot x = \frac{-sin ^2x - \cos ^2 x}{\sin ^2 x} \\ \frac{d}{dx} \cot x = -\frac{\sin ^2 x + \cos ^2 x}{\sin ^2 x} \\ \frac{d}{dx} \cot x = -\frac{1}{\sin ^2 x} \\ \frac{d}{dx} \cot x = - \csc ^2 x \quad \blacksquare \end{align}

Now let's look at an example of applying theorem 1.

## Example 1

Find the derivative of $f(x) = \sec x \csc x$.

We note that $\frac{d}{dx} \sec x = \sec x \tan x$ and $\frac{d}{dx} \csc x = -\csc x \cot x$. Applying the product rule we obtain:

(4)
\begin{align} \frac{d}{dx} f(x) = \csc x (\sec x \tan x) - \sec x (-csc x \cot x) \\ \frac{d}{dx} f(x) = \csc x \sec x \tan x - \sec x \csc x \cot x \end{align}