## Definition

A mutual direct reciprocal relation between trigonometric ratios tangent and cotangent is called reciprocal trigonometric identity of tangent and cotangent.

In trigonometry, tangent and cotangent both trigonometric ratios are defined by the lengths of opposite side and adjacent side of a right angled triangle at an angle but definitions of them are in reciprocal form. Hence, tangent and cotangent form mutual direct relation between them.

## Proof

Construct a right angled triangle and it is assumed that the angle of right angled triangle is theta $(\theta)$.

According to definition of tangent, it is expressed as ratio of length of opposite side to length of adjacent side at the respective angle.

$$\tan \theta = \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}$$

As per definition of cotangent, trigonometric function cotangent is written as ratio of length of adjacent side to length of opposite side at the respective angle.

$$\cot \theta = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}$$

1

#### Reciprocal Relation of Tangent with Cotangent

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

$$\tan \theta = \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side} = \frac{1}{\frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}}$$

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

$$\Rightarrow \tan \theta = \frac{1}{\frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}}$$

$$\Rightarrow \tan \theta = \frac{1}{\cot \theta}$$

It is proved that the reciprocal of cotangent is tangent.

2

#### Reciprocal Relation of Cotangent with Tangent

The ratio of length of adjacent side to length of opposite side can also be written in its reciprocal form.

$$\cot \theta = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side} = \frac{1}{\frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}}$$

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

$$\Rightarrow \cot \theta = \frac{1}{\frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}}$$

$$\Rightarrow \cot \theta = \frac{1}{\tan \theta}$$

It is proved that the reciprocal of tangent is cotangent.