Secant, Cosecant & Cotangent

The reciprocal trig functions extend $\sin$, $\cos$ and $\tan$, and they bring two extra Pythagorean identities that appear throughout 0606 identity and equation work.

The three definitions

$\sec\theta = \frac{1}{\cos\theta}, \qquad \operatorname{cosec}\theta = \frac{1}{\sin\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

A memory aid for the mismatched names: secant pairs with cosine, cosecant with sine, the third letter tells you the reciprocal.

The two new identities

Dividing the base identity $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ and by $\sin^2\theta$ gives: $1 + \tan^2\theta = \sec^2\theta \qquad \text{and} \qquad 1 + \cot^2\theta = \operatorname{cosec}^2\theta$

These are as useful as the original Pythagorean identity, they let you convert between $\tan$ and $\sec$, or $\cot$ and $\operatorname{cosec}$, in proofs and equations.

A worked example

Solve $\sec^2\theta = 3 + \tan\theta$ for $0^\circ \le \theta \le 180^\circ$. Use $\sec^2\theta = 1 + \tan^2\theta$: $1 + \tan^2\theta = 3 + \tan\theta \Rightarrow \tan^2\theta - \tan\theta - 2 = 0$. Factorise: $(\tan\theta - 2)(\tan\theta + 1) = 0 \Rightarrow \tan\theta = 2$ or $\tan\theta = -1$. $\tan\theta = 2 \Rightarrow \theta = 63.4^\circ$; $\tan\theta = -1 \Rightarrow \theta = 135^\circ$ (in range).

The identity turns a $\sec$ equation into a familiar quadratic in $\tan\theta$.

Common mistakes

• Pairing the names wrongly ($\sec = 1/\sin$ is wrong; $\sec = 1/\cos$).
• Misremembering the identities as $1 - \tan^2 = \sec^2$ (it’s a plus).
• Forgetting to convert to one ratio before solving.

Full topic context: Trigonometry notes and Trig Identities.