Reciprocal Trigonometric identity of Cosine and Secant

reciprocal relation of cosine and secant

Definition

A mutual direct reciprocal relation between trigonometric ratios cosine and secant is called reciprocal trigonometric identity of cosine and secant.

Cosine and secant both trigonometric ratios are defined in trigonometry by the lengths of adjacent side and hypotenuse of a right angled triangle at an angle but definitions of both of them are in reciprocal. Due to this reason, cosine and secant form reciprocal relation between them directly.

Proof

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

right angled triangle

As per definition of cosine, this trigonometric function is expressed as ratio of length of adjacent side to length of hypotenuse at the respective angle.

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

As per definition of secant, it is expressed as ratio of length of hypotenuse to length of adjacent side at the respective angle.

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

1

Reciprocal Relation of Cosine with Secant

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

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

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

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

$$\Rightarrow \cos \theta = \frac{1}{\sec \theta}$$

It is proved that the reciprocal of secant is cosine.

2

Reciprocal Relation of Secant with Cosine

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

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

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

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

$$\Rightarrow \sec \theta = \frac{1}{\cos \theta}$$

It is proved that the reciprocal of cosine is secant.

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