# Reciprocal Trigonometric identity of Sine and Cosecant

## Definition

A mutual direct reciprocal relation between trigonometric ratios sine and cosecant is called reciprocal trigonometric identity of sine and cosecant.

Sine and cosecant both trigonometric ratios are defined by the lengths of opposite side and hypotenuse of a right angled triangle at an angle but their definitions are in reciprocal. Hence, sine and cosecant can form reciprocal relation between them.

### Proof

Consider a right angled triangle and assume the angle of right angled triangle is theta $(\theta)$.

According to definition of sine, trigonometric function sine is written in mathematics as ratio of length of opposite side to length of hypotenuse at the respective angle.

$$\sin \theta = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

According to definition of cosecant, trigonometric function cosecant is written in mathematics as ratio of length of hypotenuse to length of opposite side at the respective angle.

$$\csc \theta = \frac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}$$

1

#### Reciprocal Relation of Sine with Cosecant

The ratio of length of opposite side to length of hypotenuse can be expressed in its reciprocal form.

$$\sin \theta = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{1}{\frac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}}$$

The ratio of length of hypotenuse to length of opposite side is cosecant at an angle in a right angled triangle.

$$\Rightarrow \sin \theta = \frac{1}{\frac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}}$$

$$\Rightarrow \sin \theta = \frac{1}{\csc \theta}$$

It is proved that the reciprocal of cosecant is sine.

2

#### Reciprocal Relation of Cosecant with Sine

The ratio of length of hypotenuse to length of opposite side can also be expressed in its reciprocal form.

$$\csc \theta = \frac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side} = \frac{1}{\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}}$$

The ratio of length of opposite side to length of hypotenuse is sine at an angle in a right angled triangle.

$$\Rightarrow \csc \theta = \frac{1}{\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}}$$

$$\Rightarrow \csc \theta = \frac{1}{\sin \theta}$$

It is proved that the reciprocal of sine is cosecant.