The expression <span class="math-tex">\(\rm \dfrac{\tan A}{1-\cot A} + \dfrac{\cot A}{1-\tan A}\)</span> can be written as:
The given expression is: \[ \frac{\frac{\sin x}{\cos x}}{1 - \frac{\cos x}{\sin x}} + \frac{\frac{\cos x}{\sin x}}{1 - \frac{\sin x}{\cos x}} \] \[ = \frac{\sin^2 x}{\cos x (\sin x - \cos x)} + \frac{\cos^2 x}{\sin x (\cos x - \sin x)} \] \[ = \frac{\sin^3 x - \cos^3 x}{\sin x \cos x (\sin x - \cos x)} \] \[ = \frac{(\sin x - \cos x)(\sin^2 x + \cos^2 x + \sin x \cos x)}{\sin x \cos x (\sin x - \cos x)} \] \[ = 1 + \sec x \csc x \]
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