Product identity of Sine and Cosecant

Formula

$\large \sin{\theta} \csc{\theta} \,=\, 1$

A mathematical relation between sine and cosecant functions in product form at an angle is called the product identity of sine and cosecant.

Sine and cosecant are two trigonometric functions but their definitions are in reciprocal form. Hence, they form a mathematical relation by their product for every angle and the product of them is always one for an angle.

Proof

$\Delta BAC$ is a right angled triangle and its angle is theta. Express the sine and cosecant functions in terms of ratio of lengths of the sides at this angle.

01

Expressing functions in Mathematical form

$BC$ and $AC$ are length of the opposite side and hypotenuse respectively.

According to the definition of the sine function.

$\sin{\theta} \,=\, \dfrac{BC}{AC}$

According to the definition of the cosecant function.

$\csc{\theta} \,=\, \dfrac{AC}{BC}$

02

Multiplication of the functions

Multiply both sine and co-secant functions to obtain the product of them.

$\implies \sin{\theta} \times \csc{\theta} \,=\, \dfrac{BC}{AC} \times \dfrac{AC}{BC}$

$\implies \sin{\theta} \csc{\theta} \,=\, \dfrac{BC \times AC}{AC \times BC}$

$\require{cancel} \implies \sin{\theta} \csc{\theta} \,=\, \dfrac{\cancel{BC} \times \cancel{AC}}{\cancel{AC} \times \cancel{BC}}$

$\therefore \,\,\,\,\,\, \sin{\theta} \csc{\theta} \,=\, 1$

It is proved that the product of sine and cosecant functions at an angle is one. It is called the product rule of trigonometric functions sine and cosecant in trigonometry.

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