To determine whether the statement "There is no solution to the equation [tex]\(\sec x = 0\)[/tex]" is true or false, let's carefully analyze the properties of the secant function, [tex]\(\sec x\)[/tex].
The secant function [tex]\(\sec x\)[/tex] is defined as the reciprocal of the cosine function, [tex]\(\cos x\)[/tex]. Mathematically, this relation is expressed as:
[tex]\[
\sec x = \frac{1}{\cos x}
\][/tex]
For the equation [tex]\(\sec x = 0\)[/tex] to hold true, the right-hand side of the equation must equal zero:
[tex]\[
\frac{1}{\cos x} = 0
\][/tex]
This implies that the reciprocal of [tex]\(\cos x\)[/tex] should be zero. However, for a reciprocal (or fraction) to be zero, the numerator must be zero and the denominator must be non-zero. Our equation implies that:
[tex]\[
1 = 0 \cdot \cos x
\][/tex]
Since the numerator here is a constant (1) and can never be zero, the cosine function [tex]\(\cos x\)[/tex] cannot produce a scenario where [tex]\(\frac{1}{\cos x} = 0\)[/tex]. Essentially, there is no value of [tex]\(x\)[/tex] for which [tex]\(\cos x\)[/tex] can make [tex]\(\sec x\)[/tex] equal to zero.
Therefore, it follows that the equation [tex]\(\sec x = 0\)[/tex] has no solutions, confirming that it is impossible for [tex]\(\sec x\)[/tex] to be zero.
Given this detailed reasoning, the correct answer is:
A. True
