Reciprocal identity of Tangent and Cotangent

The reciprocal relation of tangent function with cotangent function and cotangent function with tangent function is called the reciprocal identity of tangent and cotangent functions.

Formula

$(1)\,\,\,\,\,\,$ $\tan{\theta} \,=\, \dfrac{1}{\cot{\theta}}$

$(2)\,\,\,\,\,\,$ $\cot{\theta} \,=\, \dfrac{1}{\tan{\theta}}$

Introduction

Tangent and Cotangent functions are trigonometric functions but they both have reciprocal relation mutually. Therefore, tangent function is written in terms of cotangent function and cotangent function is also written in terms of tangent in reciprocal form. The reciprocal relation between tangent and cotangent functions is used as a formula in trigonometric mathematics.

Proof

Consider a right angled triangle ($\Delta BAC$) and its angle is taken as theta ($\theta$) for proving the reciprocal relation between tangent and cotangent functions in trigonometry mathematically.

right angled triangle
Define Tangent function

Write tangent function in terms of ratio of the lengths of the sides of the right angled triangle.

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

Express it in Reciprocal form

The ratio of sides for the tangent function can be written in reciprocal form in the following way.

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

Express it in terms of Cotangent

According to Trigonometry, the ratio of length of the adjacent side to length of the opposite side is called cotangent function.

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

Therefore, tangent function can be written in terms of cotangent function according to this law..

$\,\,\, \therefore \,\,\,\,\,\, \tan{\theta} \,=\, \dfrac{1}{\cot{\theta}}$

It expresses how tangent function has relation with cotangent function in reciprocal form.

The reciprocal relation of cotangent function with tangent function can also be derived in trigonometric mathematics by repeating the process one more time in the similar way.

Define Cotangent function

Write the cotangent function in terms of ratio of the lengths of the sides for the right angled triangle.

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

Write it in Reciprocal form

Express the ratio of sides of cotangent function in reciprocal form.

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

Write it in terms of Tangent

Actually, the ratio of length of opposite side to length of adjacent side is known as tangent function.

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

Therefore, it has proved that cotangent function can be expressed in terms of tangent function on the basis of this law.

$\,\,\, \therefore \,\,\,\,\,\, \cot{\theta} \,=\, \dfrac{1}{\tan{\theta}}$

Therefore, the reciprocal relation of cotangent function with tangent function and also tangent with cotangent function are used as trigonometric formulas in trigonometric mathematics.



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