The sum of the subtangent and subnormal to the curve 

$\mathrm{x}=\mathrm{c}[2 \mathrm{cos\theta }-\log (\cos \mathrm{ec\theta }+\mathrm{cot\theta }\left)\right], \mathrm{y}=\mathrm{c} \sin 2\mathrm{\theta }$ at $\mathrm{\theta }=\frac{\mathrm{\pi }}{3}$ is

$$\begin{gathered} \frac{\mathrm{dx}}{\mathrm{d\theta }}=\mathrm{c}(-2\mathrm{sin\theta }+\mathrm{cosec\theta }) \\ =\mathrm{c}\left(\frac{\cos 2\mathrm{\theta }}{\mathrm{sin\theta }}\right) \\ \frac{\mathrm{dy}}{\mathrm{d\theta }}=\mathrm{c} \cos 2\mathrm{\theta }·2 \\ \frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{sin\theta }=2\times \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=\sqrt{3} \\ \mathrm{L}.\mathrm{S}.\mathrm{T}=\frac{{\mathrm{y}}_1}{\mathrm{m}}=\frac{\mathrm{c} {\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\sqrt{3}}=\raisebox{1ex}{$\mathrm{c}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \mathrm{L}.\mathrm{S}.\mathrm{N}={\mathrm{y}}_1·\mathrm{m}=\mathrm{c}\frac{\sqrt{3}}{2}·\sqrt{3}=\frac{3\mathrm{c}}{2} \\ \mathrm{L}.\mathrm{S}.\mathrm{T}+\mathrm{L}.\mathrm{S}.\mathrm{N}=\frac{\mathrm{c}}{2}+\frac{3\mathrm{c}}{2}=2\mathrm{c} \end{gathered}$$