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

Step 1: Apply Sum-to-Product Formulas

To simplify the given expression, we will use the sum-to-product trigonometric identities. The numerator involves the sum of two cosines, and the denominator involves the difference of two sines. We apply the respective formulas to transform these sums and differences into products.

Step 2: Substitute into the Expression

Now, we substitute $A = 7x$ and $B = 5x$ into the sum-to-product formulas. This transforms the original fraction into a new form where the numerator and denominator are products of trigonometric functions.

Step 3: Simplify the Angles

We perform the arithmetic operations within the arguments of the trigonometric functions. This simplifies the angles to $6x$ and $x$, making the expression easier to work with.

Step 4: Cancel Common Terms

We observe that $2 \cos(6x)$ is a common factor in both the numerator and the denominator. Assuming $\cos(6x) \neq 0$, we can cancel this common term, which significantly simplifies the expression.

Step 5: Apply Cotangent Identity

Finally, we use the fundamental trigonometric identity that states $\frac{\cos \theta}{\sin \theta} = \cot \theta$. Applying this identity to our simplified expression, we arrive at the desired result.

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