The function f(sina)+f(cosa) does not depend on "a" only sina and cosa both appear as sin^2a and cos^2a. At that time we can apply the identity sin^2x+cos^2x=1. Otherwise sina and cosa both depend on a and therefore the function f(sina)+f(cosa).
The expression "f(sin(a)) + f(cos(a))" is the sum of the value of the function f(x) for x = sin(a) and x = cos(a). The value of sin(a) and cos(a) is also not the same but dependent on what the value of a is. If the value of f(sin(a)) + f(cos(a)) does not depend on a it is a constant. This is true only for a few functions f(x). For example if f(x) = x^2, it can be said that f(sin(a)) + f(cos(a)) is not dependent on a.
Unless the function f(x) is known it is not possible to show that f(sin(a)) + f(cos(a)) is independent of a.