You can use the code below to choose a value for l that results in the least difference between the actual data and the output defined by f. (Also, you should avoid the use of the lowercase L as a variable name, because it looks very similar to a 1 or I (one, uppercase i))
In future, you should use the {}Code button to make sure your code in ready to copy and paste to Matlab.
data=[0 0;
0.05 1.108646244630E-01;
0.10 2.217423074817E-01;
0.15 3.325947375398E-01;
0.20 4.434863433851E-01;
0.25 5.543595496420E-01;
0.30 6.652338361973E-01;
0.35 7.761094191116E-01;
0.40 8.869865144820E-01;
0.45 9.978653384221E-01;
0.50 1.108746107036E+00];
% Objective function
k=0.3;%fixed
f=@(l,x) cos(k*l)^2.*x;% function to fit
x=data(:,1);
yx=data(:,2);
intial_b_vals=0.01;
% Ordinary Least Squares cost function
OLS = @(b) sum((f(b,x) - yx).^2);
opts = optimset('MaxFunEvals',50000, 'MaxIter',10000);
% Use 'fminsearch' to minimise the 'OLS' function
fitted_b = fminsearch(OLS, intial_b_vals, opts);
For me, this results in l having an optimal value of 0.
