Function Roots and Monotonicity Analysis

For the function f(x)=(cos‌x)−x+1,x∈ℝ, between the following two statements (S1) f(x)=0 for only one value of x is [0,π].
(S2) f(x) is decreasing in [0,‌

π

2

] and increasing in [‌

π

2

,π].