Note: the text provides two opposite-sign definitions for RDabc:
11.8: where the rhs positive-sign term's covariant subscripts abc are in the same order as the lhs equation
12.7: where the rhs negative-sign term's covariant subscripts abc are in the same order as the lhs equation
Thus I don't think I can claim there is a unique proper sign for these answers; I'm using 12.7
Additionally I'm going to use Mathematica, but I'll record intermediate results


gij={{1,0},{0,Sin[a]^2}}

gijk=ConstantArray[0,{2,2,2}];
Do[ Do[ Do[
gijk[[i,j,k]]=If[k==1,D[gij[[i,j]],a],D[gij[[i,j]],b]],{k,2}],{j,2}],{i,2}]

MatrixForm[gijk[[All,All,1]]]                                           

Out[4]//MatrixForm= 0                 0

                    0                 2 Cos[a] Sin[a]


MatrixForm[gijk[[All,All,2]]]                                           

Out[5]//MatrixForm= 0   0

                    0   0


gIJ=Inverse[gij]
                           2
Out[6]= {{1, 0}, {0, Csc[a] }}

Christoffel:
Glik=ConstantArray[0,{2,2,2}];
Do[ Do[ Do[
sum=Sum[gIJ[[l,j]] (gijk[[i,j,k]]+gijk[[j,k,i]]-gijk[[k,i,j]]),{j,2}];
Glik[[l,i,k]]=sum/2,
{k,2}],{l,2}],{i,2}]

Glik[[1,All,All]]                                                       

Out[9]= {{0, 0}, {0, -(Cos[a] Sin[a])}}

Glik[[2,All,All]]                                                      

Out[10]= {{0, Cot[a]}, {Cot[a], 0}}

i.e,:
{1/22}=-(Cos[a] Sin[a])
{2/12}={2/21}=Cot[a]


Glijk=ConstantArray[0,{2,2,2,2}];
Do[ Do[ Do[ Do [
Glijk[[l,i,j,k]]=Switch[k,1,D[Glik[[l,i,j]],a],2,D[Glik[[l,i,j]],b] ],{k,2}],{l,2}],{j,2}],{i,2}]

RDabc=ConstantArray[0,{2,2,2,2}];

Do[ Do[ Do[ Do [
sum1=Sum[Glik[[j,a1,b1]] Glik[[d1,c1,j]]-Glik[[j,a1,c1]] Glik[[d1,b1,j]] ,{j,2}];
RDabc[[d1,a1,b1,c1]]=-FullSimplify[sum1+Glijk[[d1,a1,b1,c1]]-Glijk[[d1,a1,c1,b1]] ],{a1,2}],{b1,2}],{c1,2}],{d1,2}]

RDabc[[1,All,All,All]]                                                 
                                       2           2
Out[15]= {{{0, 0}, {0, 0}}, {{0, Sin[a] }, {-Sin[a] , 0}}}
RDabc[[2,All,All,All]]                                                 

Out[16]= {{{0, -1}, {1, 0}}, {{0, 0}, {0, 0}}}

i.e.,
RDabc[[1,2,1,2]]= Sin[a]^2 = -RDabc[[1,2,2,1]]
RDabc[[2,1,1,2]]=-1 = -RDabc[[2,1,2,1]]


Rab=ConstantArray[0,{2,2}];
Do[ Do [
sum1=Sum[RDabc[[j,a1,b1,j]],{j,2}];
Rab[[a1,b1]]=FullSimplify[sum1],{a1,2}],{b1,2}]
Rab                                                                                                        
                              2
Out[20]= {{-1, 0}, {0, -Sin[a] }}

RAb=ConstantArray[0,{2,2}];
Do[ Do [
sum1=Sum[gIJ[[a1,j]] Rab[[j,b1]],{j,2}];
RAb[[a1,b1]]=FullSimplify[sum1],{a1,2}],{b1,2}]

R=Tr[RAb]

Out[23]= -2