Writing t for theta, tan(t)=sin(t)sec(t),
so if t≠0 we can write: (tan(t)-sin(t))/(sin(t))^3=(sec(t)-1)/(sin(t))^2.
The expansion of cos(t)=1-t^2/2 when t is small, and sin(t)=t.
Also, sec(t)=1/cos(t)=(1-t^2/2)^-1=1+t^2/2 when t is small. Therefore sec(t)-1=t^2/2 And (sin(t))^2=t^2.
So the quotient approaches t^2/2t^2=1/2 as t approaches zero.
The limit is therefore 1/2.