Let's analyze each of the provided mathematical statements related to trigonometric identities to determine their validity.
### a. [tex]\(\cot 82 = \frac{12}{a}\)[/tex]
[tex]\(\cot(82^\circ)\)[/tex] is the cotangent of 82 degrees, which is defined as the reciprocal of [tex]\(\tan(82^\circ)\)[/tex]. Specifically:
[tex]\[ \cot(82^\circ) = \frac{1}{\tan(82^\circ)} \][/tex]
Without additional context or specific values for [tex]\(a\)[/tex], there is no standard trigonometric identity that would simplify [tex]\(\cot(82^\circ)\)[/tex] to [tex]\(\frac{12}{a}\)[/tex]. Therefore, this statement cannot be validated as true.
### b. All are true
To conclude that all statements are true, each one must be individually verified. If any statement is found to be false, this option cannot be correct.
### c. [tex]\(\sec A = \frac{c}{12}\)[/tex]
[tex]\(\sec(A)\)[/tex] is defined as the reciprocal of [tex]\(\cos(A)\)[/tex]:
[tex]\[ \sec(A) = \frac{1}{\cos(A)} \][/tex]
Again, without additional context or specific values for [tex]\(c\)[/tex] and [tex]\(A\)[/tex], this statement does not hold true in general without more information or defined relationships. Therefore, it cannot be accepted as universally true.
### d. [tex]\(\tan 18 = \frac{12}{a}\)[/tex]
The tangent of 18 degrees, [tex]\(\tan(18^\circ)\)[/tex], is a specific trigonometric value. Without any given context or specific values for [tex]\(a\)[/tex], there is no direct or known relationship that equates [tex]\(\tan(18^\circ)\)[/tex] to [tex]\(\frac{12}{a}\)[/tex]. Hence, this statement cannot be validated as true.
### e. [tex]\(\cos 18 = \frac{12}{c}\)[/tex]
[tex]\(\cos(18^\circ)\)[/tex] is a fixed trigonometric value known in trigonometry. Without supplementary context or specific values for [tex]\(c\)[/tex], there isn't a defined identity or relationship that directly equates [tex]\(\cos(18^\circ)\)[/tex] to [tex]\(\frac{12}{c}\)[/tex]. As a result, this statement cannot be affirmed as true.
### f. None are true
Given the analysis for each of the statements a through e, we found that none of them can be validated as true based on standard trigonometric definitions and without additional context or specified values.
Thus, we conclude that the correct answer is:
[tex]\[ \boxed{f. \text{None are true}} \][/tex]
This determination is made after carefully verifying each provided relationship and recognizing that they do not hold true under standard trigonometric understanding or without additional context.
