Quotient identities
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Formulas
$(1).\,\,\,\,\,\,$ $\dfrac{\sin{\theta}}{\cos{\theta}} \,=\, \tan{\theta}$
$(2).\,\,\,\,\,\,$ $\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \cot{\theta}$
A mathematical relation of two trigonometric functions in quotient form with another trigonometric function is called the quotient trigonometric identity.
Introduction
A trigonometric function is appeared with another trigonometric function in division form in some cases but it is not always possible to divide a trigonometric function by another trigonometric function. However, there are two possible cases in which the quotient of two trigonometric functions is also a trigonometric function.
The two possible cases are used as formulas in trigonometry. They are called the quotient trigonometric identities and simply called as quotient identities.
When the angle of a right triangle is represented by theta. The sine, cosine, tangent and cotangent functions are written as $\sin{\theta}$, $\cos{\theta}$, $\tan{\theta}$ and $\cot{\theta}$ respectively.
Sine by Cosine Identity
The quotient of sine by cosine at an angle is equal to the tangent at that angle.
$\dfrac{\sin{\theta}}{\cos{\theta}} \,=\, \tan{\theta}$
Cosine by Sine Identity
The quotient of cosine by sine at an angle is equal to the cotangent of that angle.
$\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \cot{\theta}$
