Prove that $\displaystyle (\sin \theta+\cos \theta)(\tan \theta+\cot \theta)=\sec \theta+\operatorname{cosec} \theta$.

\[\begin{aligned} \text { LHS }=(\sin \theta+\cos \theta) & \left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right) \\ & =(\sin \theta+\cos \theta)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta \sin \theta}\right) \\ & =\frac{(\sin \theta+\cos \theta) \cdot(1)}{\cos \theta \sin \theta} \\ & =\sec \theta+\operatorname{cosec} \theta=\mathrm{RHS} \end{aligned} \]