Sec of negative angle

A mathematical relation of secant of negative angle with secant of positive angle is called sec of negative angle identity.

Formula

$\sec{(-\theta)} \,=\, \sec{\theta}$

Proof

Geometrically, the mathematical relation between secant of negative angle and secant of positive angle can be derived mathematically in trigonometry.

Construction of triangle with positive angle

A right angled triangle, called $\Delta POQ$ is constructed with a positive angle theta and secant of angle can be expressed in mathematical form in terms of ratio of lengths of the respective sides.

$\sec{\theta} \,=\, \dfrac{OP}{OQ}$

The right angle triangle $\Delta POQ$ is actually constructed in first quadrant. So, the lengths of adjacent side and opposite side are positive, and assumed to represent them as $x$ and $y$ respectively.

$\implies$ $\sec{\theta} \,=\, \dfrac{\sqrt{x^2+y^2}}{x}$

Construction of triangle with negative angle

In the same way, construct same right angled triangle but with negative angle. Therefore, the angle of the triangle will be $–\theta$ geometrically.

Now, write trigonometric ratio secant of angle in terms of ratio of lengths of adjacent side and hypotenuse.

$\sec{(-\theta)} \,=\, \dfrac{OR}{OQ}$

In fact, the length of opposite side is $–y$ but the length of adjacent side is same due to construction of triangle with negative angle.

$\implies$ $\sec{(-\theta)} \,=\, \dfrac{\sqrt{x^2+y^2}}{x}$

Comparing Cosine functions

In order to obtain relation between them, compare both equations of secant of positive angle and secant of negative angle.

$\sec{\theta} \,=\, \dfrac{\sqrt{x^2+y^2}}{x}$

$\sec{(-\theta)} \,=\, \dfrac{\sqrt{x^2+y^2}}{x}$

The two equations clearly discloses that secant of negative angle is always equal to secant of angle.

$\,\,\, \therefore \,\,\,\,\,\,$ $\sec{(-\theta)} \,=\, \sec{\theta}$

This trigonometric identity is called secant of negative angle identity and used as a formula sometimes in trigonometric mathematics.