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Cosecant, Secant and Cotangent

Definition

The trigonometric functions, cosecant, secant and cotangent are the reciprocals of the trigonometric functions sine, cosine, tangent, respectively. \[\mathrel{\text{cosec}} \theta = \dfrac{1}{\sin \theta} \qquad\quad \sec \theta = \dfrac{1}{\cos \theta} \qquad\quad \cot \theta = \dfrac{1}{\tan \theta}\]

Identities

The most useful identities are \begin{align} 1 + \cot^2 \theta &= \mathrel{\text{cosec}}^2 \theta \\ \tan^2 \theta + 1 &= \sec^2 \theta \end{align}

Graphs

Cosecant
Secant
Cotangent

Derivatives

The derivatives of these trig functions are

$f(x)$

$f'(x)$

$\mathrel{\text{cosec}} x$

$-\cot x \mathrel{\text{cosec}} x$

$\sec x$

$\tan x \sec x$

$\cot x$

$-\mathrel{\text{cosec}}^2 x$

Worked Examples

Example 1

Show that the equation $\dfrac{1}{1+\cos\theta} + \dfrac{1}{1-\cos\theta} = 32$ can be written in the form $\mathrel{\text{cosec

^2\theta =16$. }}

Solution

First combine the two fractions.

\begin{align} \dfrac{1}{1+\cos\theta} + \dfrac{1}{1-\cos\theta} &= 32\\\\ \dfrac{(1-\cos\theta)+(1+\cos\theta)}{(1+\cos\theta)(1-\cos\theta)} &=32\\ \end{align}

Simplify.

\begin{align} \dfrac{1+1-\cos\theta+\cos\theta}{1-\cos\theta+\cos\theta-\cos^2\theta} &=32\\ \dfrac{2}{1-\cos^2\theta} &= 32\\ \dfrac{1}{1-\cos^2\theta} &= 16\\ \end{align}

Use the identity $\sin^2 \theta = 1 - \cos^2 \theta$.

\begin{align} \frac{1}{\sin^2\theta} &= 16\\\\ \mathrel{\text{cosec

^2\theta &= 16 \end{align}

}}

Example 2

Show that the equation $\dfrac{\mathrel{\text{cosec

x}{1+\mathrel{\text{cosec}} x} - \dfrac{\mathrel{\text{cosec}} x}{1-\mathrel{\text{cosec}} x}=50$ can be written in the form $\sec^2x=25$.

}}

Solution

First, combine the two fractions.

\begin{align} \frac{\mathrel{\text{cosec

x}{1+\mathrel{\text{cosec}} x} - \dfrac{\mathrel{\text{cosec}} x}{1-\mathrel{\text{cosec}} x} &=50\\\\ \frac{\mathrel{\text{cosec}} x \cdot (1-\mathrel{\text{cosec}} x) - \mathrel{\text{cosec}} x \cdot (1+\mathrel{\text{cosec}} x)}{(1+\mathrel{\text{cosec}} x)(1-\mathrel{\text{cosec}} x)} &= 50\\ \end{align}

Simplify.

\begin{align} \frac{\mathrel{\text{cosec}} x-\mathrel{\text{cosec}}^2x-\mathrel{\text{cosec}} x -\mathrel{\text{cosec}}^2 x}{1-\mathrel{\text{cosec}} x+\mathrel{\text{cosec}} x-\mathrel{\text{cosec}}^2x} &= 50\\ \frac{-2\mathrel{\text{cosec}}^2x}{1-\mathrel{\text{cosec}}^2x} &= 50\\ \frac{-\mathrel{\text{cosec}}^2x}{1-\mathrel{\text{cosec}}^2x} &=25 \end{align}

Rewrite using identities.

\begin{align} \frac{-(1+\cot^2x)}{-\cot^2x} &=25 & (\mathrel{\text{cosec}}^2 x &= 1+ \cot^2 x )\\\\ \frac{1+\cot^2x}{\cot^2x} &=25\\\\ \frac{1}{\cot^2x} + \frac{\cot^2x}{\cot^2x} &=25\\\\ \tan^2x + 1 &=25 & (\cot x &= \frac{1}{\tan x})\\ \sec^2x &=25 \end{align}

}}

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