What Are Reciprocal Identities?

In mathematics, a reciprocal of a number is the number that, when multiplied by the original number, produces 1. For example, the reciprocal for the variable x is 1/​x​, because

\(x × \frac{1}{x} = \frac{x}{x} = 1\)

In this example, 1/​x​ is the reciprocal identity of ​x​, and vice versa. In trigonometry, either of the non-90-degree angles in a right triangle can be defined by ratios called the sine, cosine and tangent. Applying the concept of reciprocal identities, mathematicians define three more ratios. Their names are cosecant, secant and cotangent. Cosecant is the reciprocal identity of sine, secant that of cosine and cotangent that of tangent.

Consider an angle ​θ​, which is one of the two non-90-degree angles in a right triangle. If the length of the side of the triangle opposite the angle is "​b​," the length of the side adjacent to the angle and opposite the hypotenuses is "​a​" and the length of the hypotenuse is "​r​," we can define the three primary trigonometric ratios in terms of these lengths.

\(\text{sine } θ = \sin θ = \frac{b}{r} \
\,\
\text{cosine }θ = \cos θ = \frac{a}{r} \
\,\
\text{tangent }θ = \tan θ = \frac{b}{a}\)

The reciprocal identity of sin ​θ​ must be equal to 1/sin θ, since that is the number that, when multiplied by sin ​θ​, produces 1. The same is true for cos ​θ​ and tan ​θ​. Mathematicians give these reciprocals the names cosecant, secant and cotangent respectively. By definition:

\(\text{cosecant }θ = \csc θ = \frac{1}{\sin θ} \
\,\
\text{secant }θ = \sec θ = \frac{1}{\cos θ} \
\,\
\text{cotangent }θ = \cot θ = \frac{1}{\tan θ}\)

You can define these reciprocal identities in terms of the lengths of the sides of the right triangle as follows:

\(\csc θ = \frac{r}{b} \
\,\
\sec θ = \frac{r}{a} \
\,\
\cot θ = \frac{a}{b}\)

The following relationships are true for any angle ​θ​:

\(\sin θ × \csc θ = 1 \
\cos θ × \sec θ = 1 \
\tan θ × \cot θ = 1\)

If you know the sine and cosine of an angle, you can derive the tangent. This is true because

\(\sin θ = \frac{b}{r} \text{ and } \cos θ = \frac{a}{r} \text{, so } \frac{\sin θ}{\cos θ} = \frac{b}{r} × \frac{r}{a} = \frac{b}{a}\)

Since this is the definition of tan θ, the following identity, known as the quotient identity, follows:

\(\frac{\sin θ}{\cos θ} = \tan θ \
\,\
\frac{\cos θ}{\sin θ} = \cot θ\)

The Pythagorean identity follows from the fact that, for any right triangle with sides ​a​ and ​b​ and hypotenuse ​r​, the following is true: ​a​² + ​b​² = ​r​² . Rearranging terms and defining ratios in terms of sine and cosine, you arrive at the following expression:

\(\sin^2 θ + \cos^2 θ = 1\)

Two other important relationships follow when you insert reciprocal identities for sine and cosine in the above expression:

\(\tan^2 θ + 1 = \sec^2 θ \
\cot^2 θ + 1 = \csc^2 θ\)