$VT=\dfrac{\tan \alpha}{\sin\alpha}-\dfrac{\sin\alpha}{\cot\alpha}\\ =\dfrac{\dfrac{\sin\alpha}{\cos\alpha}}{\sin\alpha}-\dfrac{\sin\alpha}{\dfrac{\cos\alpha}{\sin\alpha}}\\ =\dfrac{1}{\cos\alpha}-\dfrac{\sin^2\alpha}{\cos\alpha}\\ =\dfrac{1-\sin^2\alpha}{\cos\alpha}\\ =\dfrac{\cos^2\alpha}{\cos\alpha}\\ =\cos\alpha\\ =VT$