Math Doubts

Reciprocal identity of Cosine and Secant

The reciprocal relation of cosine function with secant function and secant function with cosine is called the reciprocal identity of cosine and secant functions.

Formula

$(1)\,\,\,\,\,\,$ $\cos{\theta} \,=\, \dfrac{1}{\sec{\theta}}$

$(2)\,\,\,\,\,\,$ $\sec{\theta} \,=\, \dfrac{1}{\cos{\theta}}$

Introduction

According to trigonometry, cosine and secant functions have mutual reciprocal relation. Hence, cosine function can be written in terms of secant in reciprocal form directly and also secant function can also be written in terms of cosine function in reciprocal form. The reciprocal relationship between cosine and secant functions is used as a trigonometric formula in mathematics.

Proof

Take a right angled triangle ($\Delta BAC$), whose angle is considered as theta ($\theta$) for proving the reciprocal relation between cosine and secant functions mathematically.

right angled triangle
Define Cosine function

Express the cosine function in terms of ratio of the lengths of the sides of a right angled triangle.

$\cos{\theta} \,=\, \dfrac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$

Express it in Reciprocal form

The ratio of sides for the cosine function can be written in reciprocal form as follows.

$\cos{\theta} \,=\, \dfrac{1}{\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}}$

Express it in terms of Secant

As per Trigonometry, the ratio of length of the hypotenuse to length of the adjacent side is called Secant function.

$\sec{\theta} \,=\, \dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}$

Therefore, cosine function can be written in terms of Secant function as per this rule.

$\,\,\, \therefore \,\,\,\,\,\, \cos{\theta} \,=\, \dfrac{1}{\sec{\theta}}$

This trigonometric identity represents how cosine function has relation with secant function in reciprocal form.

In this way, the reciprocal relation of secant function with cosine function can also be derived in mathematics by repeating the procedure one more time to prove it in trigonometry.

Defining Secant function

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

$\sec{\theta} \,=\, \dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}$

Write it in Reciprocal form

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

$\sec{\theta} \,=\, \dfrac{1}{\dfrac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}}$

Write it in terms of Cosine

In fact, the ratio of length of adjacent side to length of the hypotenuse is known as cosine function.

$\cos{\theta} \,=\, \dfrac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$

Therefore, it is proved that secant function can be expressed in terms of cosine function on the basis of this law.

$\,\,\, \therefore \,\,\,\,\,\, \sec{\theta} \,=\, \dfrac{1}{\cos{\theta}}$

Therefore, the reciprocal relation of cosine with secant and secant with cosine functions are used as formulas to replace cosine by secant and vice-versa in trigonometric mathematics.



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