$VT=\dfrac{\tan\alpha}{1+\cot\alpha}+\dfrac{\cot\alpha}{1+\tan\alpha}$
$=\dfrac{\dfrac{\sin\alpha}{\cos\alpha} }{ 1+\dfrac{\cos\alpha}{\sin\alpha} }+\dfrac{ \dfrac{\cos\alpha}{\sin\alpha} }{1+\dfrac{\sin\alpha}{\cos\alpha}}$
$=\dfrac{\sin\alpha}{\cos\alpha}.\dfrac{\sin\alpha}{\sin\alpha+\cos\alpha}+\dfrac{\cos\alpha}{\sin\alpha}.\dfrac{\cos\alpha}{\sin\alpha+\cos\alpha}$
$=\dfrac{1}{\sin\alpha+\cos\alpha}\Big( \dfrac{\sin^2\alpha}{\cos\alpha}+\dfrac{\cos^2\alpha}{\sin\alpha}\Big)$
$=\dfrac{1}{\sin\alpha+\cos\alpha}.\dfrac{\sin^3\alpha+\cos^3\alpha}{\sin\alpha\cos\alpha}$
$=\dfrac{\sin^2\alpha+\cos^2\alpha-\sin\alpha\cos\alpha}{\sin\alpha\cos\alpha}$
$=\dfrac{\sin^2\alpha}{\sin\alpha\cos\alpha}+\dfrac{\cos^2\alpha}{\sin\alpha+\cos\alpha}-1$
$=\tan\alpha+\cot\alpha-1$
$=VP$
