The C programming cosh function is defined in the math.h header file and is used to calculate the hyperbolic cosine of a given value x.

The hyperbolic cosine function, denoted cosh(x), is defined as:

cosh(x) = (e^x + e^(-x)) / 2

where e is the base of the natural logarithm.

The cosh function takes a single argument of type double, which represents the value whose hyperbolic cosine is to be calculated. The function returns a double value, which represents the calculated hyperbolic cosine.

Here's an example usage of the cosh function to calculate the hyperbolic cosine of a given input:

#include <stdio.h>
#include <math.h>

int main() {
    double x = 0.5;
    double hyperbolic_cosine = cosh(x);
    
    printf("The hyperbolic cosine of %f is %f\n", x, hyperbolic_cosine);
    
    return 0;
}

In this example, the cosh function is used to calculate the hyperbolic cosine of 0.5, which is stored in the x variable. The calculated hyperbolic cosine is then stored in the hyperbolic_cosine variable, and printed to the console using the printf function.

Note that the cosh function is only defined for real values of x. Also note that the hyperbolic cosine function grows very quickly for large positive or negative values of x, so it is important to be aware of potential overflow issues when working with large values.