$(1+\tan \theta+\cot \theta)(\sin \theta-\cos \theta)=\left(\frac{\sec \theta}{\operatorname{cosec}^{2} \theta}-\frac{\operatorname{cosec} \theta}{\sec ^{2} \theta}\right)$
$\mathrm{LHS}=(1+\tan \theta+\cot \theta)(\sin \theta-\cos \theta)$
$=\sin \theta+\tan \theta \sin \theta+\cot \theta \sin \theta-\cos \theta-\tan \theta \cos \theta-\cot \theta \cos \theta$
$=\sin \theta+\tan \theta \sin \theta+\frac{\cos \theta}{\sin \theta} \times \sin \theta-\cos \theta-\frac{\sin \theta}{\cos \theta} \times \cos \theta-\cot \theta \cos \theta$
$=\sin \theta+\tan \theta \sin \theta+\cos \theta-\cos \theta-\sin \theta-\cot \theta \cos \theta$
$=\tan \theta \sin \theta-\cot \theta \cos \theta$
$=\frac{\sin \theta}{\cos \theta} \times \frac{1}{\operatorname{cosec} \theta}-\frac{\cos \theta}{\sin \theta} \times \frac{1}{\sec \theta}$
$=\frac{1}{\operatorname{cosec} \theta} \times \frac{1}{\operatorname{cosec} \theta} \times \sec \theta-\frac{1}{\sec \theta} \times \frac{1}{\sec \theta} \times \operatorname{cosec} \theta$
$=\frac{\sec \theta}{\operatorname{cosec}^{2} \theta}-\frac{\operatorname{cosec} \theta}{\sec ^{2} \theta}$
$=\mathrm{RHS}$
Hence, LHS $=$ RHS