Reciprocal and Quotient Identities (HSC SSCE Mathematics Advanced): Revision Notes

Reciprocal and Quotient Identities

In trigonometry, we use special relationships called identities to connect different trigonometric functions. Two important types are reciprocal identities and quotient identities. These identities help us simplify expressions, solve equations, and prove trigonometric statements.

What is an identity?

An identity is a mathematical statement involving one or more variables that remains true for all possible values of those variables (except where undefined). Unlike equations that are only true for specific values, identities are always true within their domain.

Reciprocal identities

The reciprocal identities connect three less common trigonometric functions (secant, cosecant, and cotangent) to the three primary functions (cosine, sine, and tangent). The word reciprocal means we flip the fraction, so each reciprocal function equals $\frac{1}{\text{primary function}}$.

The three reciprocal identities

$\text{cosec } \theta = \frac{1}{\sin \theta}$

$\sec \theta = \frac{1}{\cos \theta}$

$\cot \theta = \frac{1}{\tan \theta}$

Where do these come from?

These identities emerge naturally from how we define trigonometric functions using the unit circle or right-angled triangles.

For example, on a unit circle, if $\cos \theta = \frac{x}{r}$ where $r$ is the radius, then $\sec \theta = \frac{r}{x}$. When $r = 1$ (unit circle), this becomes $\sec \theta = \frac{1}{x} = \frac{1}{\cos \theta}$.

When are reciprocal identities undefined?

Because reciprocal identities involve fractions, they become undefined whenever their denominator equals zero. Here's when each function is undefined:

• sec $\theta$ is undefined when $\cos \theta = 0$, which occurs at $\theta = 90° + 180°n$ (for example: $90°$, $270°$, etc.), where $n$ is any integer

• cosec $\theta$ is undefined when $\sin \theta = 0$, which occurs at $\theta = 180°n$ (for example: $0°$, $180°$, $360°$, etc.), where $n$ is any integer

• cot $\theta$ is undefined when $\tan \theta = 0$ (at $\theta = 180°n$) or when $\tan \theta$ itself is undefined (at $\theta = 90° + 180°n$), where $n$ is any integer

Quotient identities

The quotient identities express tangent and cotangent as ratios (quotients) of sine and cosine. These relationships come directly from the unit circle, where sine and cosine represent the $y$ and $x$ coordinates respectively.

The two quotient identities

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

$\cot \theta = \frac{\cos \theta}{\sin \theta}$

Understanding the quotient identities

On the unit circle, if we have a point at angle $\theta$ with coordinates $(x, y)$, then:

• $\tan \theta = \frac{y}{x}$, and since $y = \sin \theta$ and $x = \cos \theta$, we get $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• Similarly, $\cot \theta = \frac{x}{y} = \frac{\cos \theta}{\sin \theta}$

These identities are undefined when their denominators equal zero, just like the reciprocal identities.

Proving identities using reciprocal and quotient identities

When proving trigonometric identities, we typically work with one side of the equation (usually the left-hand side) and transform it step-by-step until it matches the other side. The key strategy is to substitute the reciprocal and quotient identities, then simplify algebraically.