When the string of a conical pendulum makes an angle of 45^o with the vertical, its period is T₁. When the string makes an angle of 60^o with the vertical, its time period is T₂. Then \(\frac{T^2_1}{T^2_1}\) is : 

0.5

In conical pendulum :

\(T \cos {\theta} = mg = constant\)

\(\Rightarrow T_1 \cos {\theta_1} = T_2 \cos {\theta_2}\)

\(\Rightarrow \frac{T_1}{T_2} = \frac{\cos {\theta_1}}{\cos {\theta_2}} = \frac{\cos {60^o}}{\cos {45^o}}\)

\(\Rightarrow \frac{T_1}{T_2} = \frac{1/2}{1/sqrt{2}}\)

\(\Rightarrow \frac{T_1}{T_2} = \frac{1}{\sqrt{2}}\)

\(\Rightarrow [\frac{T_1}{T_2}]^2 = \frac{1}{2}\)