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Reciprocal identity of Sec function

Formula

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

Proof

Secant is a ratio of lengths of hypotenuse to adjacent side and the cosine is a ratio of lengths of adjacent side to hypotenuse. Mutually, secant and cosine functions are reciprocals. Therefore, the reciprocal of secant of angle equals to cosine of angle.

$\Delta RPQ$ is a right triangle and its angle is denoted by theta ($\theta$).

right triangle

Write secant in its ratio form

Write the secant of angle theta ($\sec{\theta}$) in its ratio form.

$\sec{\theta} \,=\, \dfrac{PQ}{PR}$

Write cosine in its ratio form

Now, write cos of angle theta ($\cos{\theta}$) in its ratio form.

$\cos{\theta} \,=\, \dfrac{PR}{PQ}$

Relation between secant and cosine

Write the value in ratio form of cosine function in its reciprocal form and it is useful to know the relation between secant and cosine functions.

$\implies \cos{\theta} \,=\, \dfrac{1}{\dfrac{PQ}{PR}}$

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

Therefore, it has proved mathematically that the reciprocal of secant function is equal to cos function and it is used as a formula in trigonometry.

Note

In mathematics, any symbol can be used to denote the angle of right triangle but the reciprocal identity of secant function has to be written in terms of the respective angle.

For example, if angle of right triangle is expressed as $x$, then

$\cos{x} \,=\, \dfrac{1}{\sec{x}}$

Similarly, if angle of right triangle is expressed as $A$, then

$\cos{A} \,=\, \dfrac{1}{\sec{A}}$

The reciprocal identity of secant function is written in this form but the angle of the right triangle is only the changing factor in the formula.

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