\(R_1 \propto T_1 ==> tan \theta \propto kT_1 ==> tan \theta = kT_1\) ……. \((i)\)

\(R_2  \propto T_2 ==> tan (90° -\theta) ) T_2 ==> cot \theta = kT_2 \)……..\((ii)\)

\(k({T_2} - {T_1}) = (\cot \theta - \tan \theta )\,\)

\(\,({T_2} - {T_1}) = \left( {\frac{{\cos \theta }}{{\sin \theta }} - \frac{{\sin \theta }}{{\cos \theta }}} \right) = \frac{{({{\cos }^2}\theta - {{\sin }^2}\theta )}}{{\sin \theta \cos \theta }} = \frac{{\cos 2\theta }}{{\sin \theta \cos \theta }} = \cot 2\theta \)

\(==> (T_2 -T_1) )\,\propto\, cot 2\theta\)