[tex] \cot\theta-\cot2\theta=\dfrac{1}{\sin2\theta}\\\\L_s=\dfrac{\cos\theta}{\sin\theta}-\dfrac{\cos2\theta}{\sin2\theta}=\dfrac{\cos\theta}{\sin\theta}-\dfrac{\cos^2\theta-\sin^2\theta}{2\sin\theta\cos\theta}\\\\=\dfrac{\cos\theta\cdot2\cos\theta}{\sin\theta\cdot2\cos\theta}-\dfrac{cos^2\theta-\sin^2\theta}{2\sin\theta\cos\theta}=\dfrac{2\cos^2\theta}{2\sin\theta\cos\theta}-\dfrac{\cos^2\theta-\sin^2\theta}{2\sin\theta\cos\theta} [/tex]
[tex] =\dfrac{2\cos^2\theta-\cos^2\theta+\sin^2\theta}{\sin2\theta}=\dfrac{\cos^2+\sin^2\theta}{\sin2\theta}=\dfrac{1}{\sin2\theta}=R_s\\\\\text{Used:}\\\\\cot x=\dfrac{\cos x}{\sin x}\\\\\sin2x=2\sin x\cos x\\\\\sin^2x+\cos^2x=1 [/tex]
