(sec A – cos A) (cot A + tan A) = sec A tan A

$(\sec A-\cos A)(\cot A+\tan A)$

$=\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}\right)$

$=\left(\frac{1-\cos ^{2} A}{\cos A}\right)\left(\frac{\cos ^{2} A+\sin ^{2} A}{\sin A \cos A}\right)$

$=\frac{\sin ^{2} A}{\cos A} \times \frac{1}{\sin A \cos A} \quad\left(\sin ^{2} \theta+\cos ^{2} \theta=1\right)$

$=\frac{\sin A}{\cos A} \times \frac{1}{\cos A}$

$=\tan A \sec A$