The question is $R:S^1\rightarrow S^1$be rotation by $\alpha$ radians,prove that $R\simeq 1_{S}$.Then conclude that every continuous map $f:S^1\rightarrow S^1$is homotopic to a continuous map $g:S^1\rightarrow S^1$with $g(1)=1$.
The first part of this question is very easy,but I really don't know how to conclude the second part of this question. I think I need a hint to solve the second question.
Given the first part then I think the second part is quite straightforward.
Given any $g: S^1\to S^1$ let $g(1)=a$. Let $R_{-a}: S^1\to S^1$ be rotation by $(-a)$ radians. The composition, \begin{equation} R_{-a}\circ g\simeq 1_{S^1}\circ g= g. \end{equation} Moreover, $(R_{-a}\circ g)(1)= R_{-a}(g(1))= R_{-a}(a)= 1$ and the composition of two continuous maps is also continuous. Thus, $g$ is homoropic to a continuous map that takes $1$ to $1$.