$=(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)$

$=(\sin \alpha+\cos \alpha)\left(\frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}\right)$ $\left[\because \tan \theta=\frac{\sin \theta}{\cos \theta}\right.$ and $\left.\cot \theta=\frac{\cos \theta}{\sin \theta}\right]$

$=(\sin \alpha+\cos \alpha)\left(\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{\sin \alpha \cdot \cos \alpha}\right)$

$=(\sin \alpha+\cos \alpha) \cdot \frac{1}{(\sin \alpha \cdot \cos \alpha)}$ $\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$

$=\frac{1}{\cos \alpha}+\frac{1}{\sin \alpha}$ $\left[\because \sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right]$

$=\sec \alpha+\cos \alpha=\operatorname{RHS}$