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Why are cosine and secant called even functions?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In mathematics, a function is classified as an even function if it satisfies the condition $f (x) = f (- x)$ for all $x$ in its domain, meaning the function's value remains unchanged when the sign of the independent variable is reversed.12

In the context of trigonometry, the cosine and secant functions are considered even because they are symmetrical about the y-axis. When an angle $A$ is placed in a standard position on a coordinate system, a point $(x, y)$ on its terminal side corresponds to a point $(x, - y)$ on the terminal side of the negative angle $- A$ . Because the cosine is defined by the ratio involving the x-coordinate and the secant is the reciprocal of the cosine, these functions yield the same value for both $A$ and $- A$ , expressed as:3

$\cos (- A) = \cos A$ 4
$\sec (- A) = \sec A$ 5

In contrast, the other four trigonometric functions are classified as odd functions because they are symmetrical about the origin and satisfy the condition $f (- x) = - f (x)$ .3