Final answer:
The function f(x) = (cosx)/(1+sinx) is neither an odd nor an even function since it does not satisfy the symmetry conditions for either. When evaluating f(-x), the resulting expression is neither the original function nor its negative, indicating it lacks the required symmetries.
Step-by-step explanation:
To determine whether the function f(x) = \frac{\cos x}{1 + \sin x} is an even or odd function, we need to investigate if it satisfies the symmetry properties that define even and odd functions.
An even function satisfies the condition f(x) = f(-x) for all x in the domain of f, which means its graph is symmetric with respect to the y-axis. An odd function satisfies the condition f(x) = -f(-x) for all x in the domain of f, which means its graph is symmetric with respect to the origin.
Let's evaluate f(-x) for our function:
f(-x) = \frac{\cos(-x)}{1 + \sin(-x)}
Since cosine is an even function, \cos(-x) = \cos(x), and since sine is an odd function, \sin(-x) = -\sin(x), we get:
f(-x) = \frac{\cos(x)}{1 - \sin(x)}
This expression is not equal to f(x) nor is it the negative of f(x); therefore, f(x) is neither an odd nor an even function.