[tex]\displaystyle
\lim_{\theta\to 0}\dfrac{1-\cos \theta}{2\sin ^2\theta}=\\\\
\lim_{\theta\to 0}\dfrac{(1-\cos \theta)(1+\cos \theta)}{2\sin ^2\theta(1+\cos \theta)}=\\\\
\lim_{\theta\to 0}\dfrac{1-\cos^2 \theta}{2\sin ^2\theta+2\sin^2 \theta\cos \theta}=\\\\
\lim_{\theta\to 0}\dfrac{\sin^2 \theta}{2\sin ^2\theta+2\sin^2 \theta\cos \theta}=\\\\
\lim_{\theta\to 0}\dfrac{1}{2+2\cos \theta}=\\\\
\dfrac{1}{2+2\cdot1}=\\\\
\dfrac{1}{4}
[/tex]
