### Instruction: Simplify $\cos 54^\circ \cos 4^\circ - \cos 36^\circ \cos 86^\circ.$

Enter your answer is a trigonometric function evaluated at an integer, such as "sin 7".  (The system will assume that the angle is in degrees.).### Response: To simplify this expression, I will use the product-to-sum formula for cosine, which states that $\cos \alpha \cos \beta = \frac{1}{2}(\cos(\alpha + \beta) + \cos(\alpha - \beta)).$
Applying this formula to each term, I get $\cos 54^\circ \cos 4^\circ - \cos 36^\circ \cos 86^\circ = \frac{1}{2}(\cos 58^\circ + \cos 50^\circ) - \frac{1}{2}(\cos 122^\circ + \cos 50^\circ).$
Simplifying, I get $\frac{1}{2}(\cos 58^\circ - \cos 122^\circ).$
Now, I notice that 58 and 122 are supplementary angles, meaning that they add up to 180 degrees.
This means that $\cos 58^\circ = -\cos 122^\circ.$
Therefore, I can rewrite the expression as $\frac{1}{2}(-2\cos 122^\circ) = -\cos 122^\circ.$
This is the simplest form of the expression as a trigonometric function evaluated at an integer.