# Derivation of Basic Identities

The derivation of basic identities can be done easily by using the functions of a right triangle. For easy reference, these trigonometric functions are listed below.

a/c = sin θ

b/c = cos θ

a/b = tan θ

c/a = csc θ

c/b = sec θ

b/a = cot θ

b/c = cos θ

a/b = tan θ

c/a = csc θ

c/b = sec θ

b/a = cot θ

**Sine and Cosecant are reciprocal to each other**

$\sin \theta = \dfrac{a}{c}$

$\sin \theta = \dfrac{a/a}{c/a}$

$\sin \theta = \dfrac{1}{\csc \theta}$

and

$\csc \theta = \dfrac{1}{\sin \theta}$

**Cosine and Secant are reciprocal to each other**

$\cos \theta = \dfrac{b}{c}$

$\cos \theta = \dfrac{b/b}{c/b}$

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

and

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

**Tangent and Cotangent are reciprocal to each other and Tangent is the ratio of Sine to Cosine**

$\tan \theta = \dfrac{a}{b}$

$\tan \theta = \dfrac{a/a}{b/a}$

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

$\tan \theta = \dfrac{a}{b}$

$\tan \theta = \dfrac{a/c}{b/c}$

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

Thus,

$\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$

and

$\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$