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The cosine of sum formula is a key trigonometric identity that allows you to simplify the cosine of the sum of two angles. It is expressed as:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \].
For instance, if you have two angles, let's name them angle X and angle Y, the formula lets you find the cosine of their sum (X+Y) by combining the cosines and sines of X and Y individually. This is incredibly useful because it breaks down complex trigonometric expressions into more manageable parts.

Putting It Into Practice

For the exercise problem \( \cos 2x \cos 9x + \sin 2x \sin 9x \), we identify that this expression resembles the right side of the cosine of sum formula, given that A is \( 2x \) and B is \( 9x \). The key is to recognize patterns that align with the formula to facilitate the simplification process.

The cosine of difference formula is essentially a twin to the cosine of sum formula, but, as its name suggests, it is used when dealing with the difference between two angles. The formula is given as:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \].
This trigonometric identity cleverly allows for the transformation of a cosine expression involving a subtraction into a sum involving individual sines and cosines.

Application in Our Problem

In our given problem, paying attention to the '+' sign in the expression \( \cos 2x \cos 9x + \sin 2x \sin 9x \) leads us to use the cosine of difference formula. By substituting A with \( 2x \) and B with \( 9x \), we match the original expression perfectly and apply the formula to rewrite the expression as \( \cos(2x - 9x) \).

Some trigonometric functions have specific properties related to symmetry. The even property of trigonometric functions tells us that for the cosine function, \( \cos(-\theta) = \cos(\theta) \). This means that the cosine function is symmetric with respect to the y-axis, or in other words, its graph looks the same on the left and right sides of the y-axis.
This property is particularly helpful when dealing with trigonometric expressions that include negative angles.

Utilizing the Even Property

In the final step of our exercise, we have reached the expression \( \cos(-7x) \). By using the even property, we simplify this to \( \cos(7x) \), removing the negative sign with confidence. This property reassures us that the cosine value of an angle does not change even if the angle is taken in the opposite direction, simplifying our calculations and understanding of trigonometric equations.