Prove that $\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \cot x$.

(N/A) Using the sum-to-product formulas:
$\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$
$\sin A - \sin B = 2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}$
Applying these to the $L.H.S.$:
$L.H.S. = \frac{2 \cos \frac{7x+5x}{2} \cos \frac{7x-5x}{2}}{2 \cos \frac{7x+5x}{2} \sin \frac{7x-5x}{2}}$
Simplifying the expression:
$= \frac{\cos \frac{12x}{2} \cos \frac{2x}{2}}{\cos \frac{12x}{2} \sin \frac{2x}{2}}$
$= \frac{\cos 6x \cos x}{\cos 6x \sin x}$
$= \frac{\cos x}{\sin x} = \cot x = R.H.S.$