To determine which function has the domain of all real numbers except [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex] where [tex]\( n \)[/tex] is an integer, we need to analyze the domains of the given functions one by one.
### A. [tex]\( y = \frac{1}{\sin x} \)[/tex]
The function [tex]\( y = \frac{1}{\sin x} \)[/tex] is defined wherever [tex]\( \sin x \)[/tex] is not zero. [tex]\( \sin x \)[/tex] is zero at [tex]\( x = n\pi \)[/tex] for integer [tex]\( n \)[/tex]. Therefore, the domain of this function is all real numbers except [tex]\( x = n\pi \)[/tex].
### B. [tex]\( y = \cot x \)[/tex]
The function [tex]\( y = \cot x \)[/tex] (which is [tex]\( \frac{\cos x}{\sin x} \)[/tex]) is defined wherever [tex]\( \sin x \)[/tex] is not zero. Like the previous case, [tex]\( \sin x \)[/tex] is zero at [tex]\( x = n\pi \)[/tex]. Thus, the domain of this function is all real numbers except [tex]\( x = n\pi \)[/tex].
### C. [tex]\( y = \sec x \)[/tex]
The function [tex]\( y = \sec x \)[/tex] (which is [tex]\( \frac{1}{\cos x} \)[/tex]) is defined wherever [tex]\( \cos x \)[/tex] is not zero. [tex]\( \cos x \)[/tex] is zero at [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex] for integer [tex]\( n \)[/tex]. Consequently, the domain of this function is all real numbers except [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex].
### D. [tex]\( y = \cos x \)[/tex]
The function [tex]\( y = \cos x \)[/tex] is defined for all real numbers since [tex]\( \cos x \)[/tex] is a continuous and periodic function without any discontinuities or undefined points. Thus, its domain is all real numbers.
### Conclusion
The function that has a domain of all real numbers except [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex] is option C, which is [tex]\( y = \sec x \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
