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.
Right triangle with sides a, b, and c and angle theta

a/c = sin 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

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

 

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