Prove that

$\mathrm{LHS}=\frac{\sin \theta}{(\cot \theta+\operatorname{cosec} \theta)}-\frac{\sin \theta}{(\cot \theta-\operatorname{cosec} \theta)}$

$=\sin \theta\left\{\frac{(\cot \theta-\operatorname{cosec} \theta)-(\cot \theta+\operatorname{cosec} \theta)}{(\cot \theta+\operatorname{cosec} \theta)(\cot \theta-\operatorname{cosec} \theta)}\right\}$

$=\sin \theta\left\{\frac{-2 \operatorname{cosec} \theta}{\left(\cot ^{2} \theta-\operatorname{cosec}^{2} \theta\right)}\right\}$

$=\sin \theta\left(\frac{-2 \operatorname{cosec} \theta}{-1}\right) \quad\left(\because \operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\right)$

$=\sin \theta .2 \operatorname{cosec} \theta$

$=\sin \theta \times 2 \times \frac{1}{\sin \theta}$

$=2$

$=$ RHS