\((\cfrac{\lambda_i}{sin(\theta_i)}+a)tan(\theta_i)=OC=\cfrac{\lambda_r}{tan(\cfrac{1}{2}\theta_r)}\)
\(\cfrac{\lambda_i}{\lambda_r}(\cfrac{1}{sin(\theta_i)}+\cfrac{a}{\lambda_i})tan(\theta_i)=\cfrac{1}{tan(\cfrac{1}{2}\theta_r)}\)
If \(\cfrac{a}{\lambda_i}\approx 1\)
\(\cfrac{\lambda_i}{\lambda_r}(sec(\theta_i)+tan(\theta_i))=\cfrac{1}{tan(\cfrac{1}{2}\theta_r)}\)
\(\cfrac{\lambda_i}{\lambda_r}(tan(\cfrac{1}{2}(\theta_i+\cfrac{\pi}{2})))=\cfrac{1}{tan(\cfrac{1}{2}\theta_r)}\)
\(\cfrac{\lambda_i}{\lambda_r}=\cfrac{1+cos(\theta_r)}{sin(\theta_r)}\cfrac{1+cos(\theta_i+\cfrac{\pi}{2})}{sin(\theta_i+\cfrac{\pi}{2})}\)
\(\cfrac{\lambda_i}{\lambda_r}=\cfrac{1+cos(\theta_r)}{sin(\theta_r)}\cfrac{1-sin(\theta_i)}{cos(\theta_i)}\)
or
\(\cfrac{\lambda_i}{\lambda_r}=\cfrac{cot(\cfrac{1}{2}(\theta_i+\cfrac{\pi}{2}))}{tan(\cfrac{1}{2}\theta_r)}\)
which is still not Snell's Law.
