In[27]:= Integrate[
DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], Pi/10, Pi}]
Out[27]= 1
and
In[37]:= Integrate[
DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], -Pi/10, +Pi/2}]
Out[37]= 1 - HeavisideTheta[0]
This integral is rather usual in Physics, for spherical coordinates.