Question:

<p><strong>Step 1:</strong> Convert to sine-cosine form. Let the expression be E.</p><p>E = $\frac{\frac{\sin A}{\cos A}}{1 - \frac{\cos A}{\sin A}} + \frac{\frac{\cos A}{\sin A}}{1 - \frac{\sin A}{\cos A}}$</p><p><strong>Step 2:</strong> Simplify denominators:</p><p>E = $\frac{\frac{\sin A}{\cos A}}{\frac{\sin A - \cos A}{\sin A}} + \frac{\frac{\cos A}{\sin A}}{\frac{\cos A - \sin A}{\cos A}}$</p><p><strong>Step 3:</strong> Simplify each fraction:</p><p>E = $\frac{\sin^2 A}{\cos A(\sin A - \cos A)} + \frac{\cos^2 A}{\sin A(\cos A - \sin A)}$</p><p><strong>Step 4:</strong> Factor out negative from second term:</p><p>E = $\frac{\sin^2 A}{\cos A(\sin A - \cos A)} - \frac{\cos^2 A}{\sin A(\sin A - \cos A)}$</p><p><strong>Step 5:</strong> Combine with common denominator $(\sin A - \cos A)$:</p><p>E = $\frac{\sin^3 A - \cos^3 A}{\sin A \cos A(\sin A - \cos A)}$</p><p><strong>Step 6:</strong> Use factorization $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$:</p><p>E = $\frac{(\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A)}{\sin A \cos A(\sin A - \cos A)}$</p><p><strong>Step 7:</strong> Cancel $(\sin A - \cos A)$ and use $\sin^2 A + \cos^2 A = 1$:</p><p>E = $\frac{1 + \sin A \cos A}{\sin A \cos A}$ = $\frac{1}{\sin A \cos A} + 1$ = $\tan A + \cot A + 1$</p><p>∴ Answer: <strong>$\tan A + \cot A + 1$ (or equivalent form)</strong></p>