Question

the function f: R->R defined by f(x)=cos(x) is 1-1

Answer 1

f:  R  ->   R     
 f(x) =  cos (x)

is not a one to one function.  It is  a many to one function.
   f(x) =  f(x+ 2 π) = f( x + 4 π) = f(x - 2π)  = f(2π - x) = f (x - 2n π)
        = f(x + 2 n π)

for example  cos 0 = 1/2 = cos 2π = cos 4π = cos (-2π) .....

Since many values of x are mapping to one value of  f(x) in the range set of f, it is a many to one function.