\({T_1}\sin {\theta _1} = - mg\)…..(ii)

\(T_1^2\left( {{{\sin }^2}{\theta _1} + {{\cos }^2}{\theta _1}} \right) = 2{\left( {mg} \right)^2}\)

\(\therefore \) \({T_1} = \sqrt 2 mg\)

\({T_2}\cos {\theta _2} = mg + {T_1}\cos {\theta _1}\)

\({T_2}\cos {\theta _2} = mg + \sqrt 2 mg\,\cos 45^\circ \)

\({T_2}\cos {\theta _2} = 2mg\)…..(i)

\({T_2}\sin {\theta _2} = {T_1}\sin {\theta _1}\)

\( = \sqrt 2 mg\,\sin 45^\circ \)

\({T_2}\sin {\theta _2} = mg\)…..(ii)

\(T_2^2 = 5{(mg)^2}\) or \({T_2} = \sqrt 5 mg\)