91Ó°ÊÓ

To understand the identity presented in the exercise, we first look at the cosine sum and difference identities. These are essential tools in trigonometry for breaking down complex trigonometric expressions into more manageable parts. The sum identity for cosine states: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\). This formula allows us to express the cosine of a sum of angles in terms of the sines and cosines of the individual angles.

Similarly, the difference identity for cosine is \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). This formula is very similar to the sum formula but with a plus sign between the sine terms, reflecting the distinct contributions of angle subtraction.

In the problem, these identities are applied to express the product \(\cos(x+y) \cos(x-y)\) in terms of \(\cos x\), \(\cos y\), \(\sin x\), and \(\sin y\). By recognizing these identities, you can simplify what might initially appear as a complex multiplication into a simpler algebraic form.

The difference of squares is a fundamental algebraic identity that comes in handy when simplifying expressions like the one in the exercise. The formula itself is: \(a^2 - b^2 = (a-b)(a+b)\). It states that subtracting one square from another can be expressed as the product of two binomials, which is particularly powerful in factoring tasks.

In the step-by-step solution, after applying the cosine sum and difference identities, we see that the expression obtained is \((\cos x \cos y)^2 - (\sin x \sin y)^2\). This form is perfect for applying the difference of squares identity, which allows us to factor the expression easily. Breaking down complex trigonometric products into something simpler via difference of squares makes it easier to subsequently apply other trigonometric or algebraic identities to reach the desired result.

By using this identity, the multiplication of two complex expressions was reduced to a simpler expression involving only \(\cos^2 x\) and \(\sin^2 y\), which was the target expression to prove further.

The Pythagorean identities are a cornerstone of trigonometric simplifications. They're based on the famous Pythagorean theorem and apply to trigonometric functions. The most common Pythagorean identity is: \(\cos^2 a + \sin^2 a = 1\), which describes the fundamental relationship between the sine and cosine of any angle.

In the exercise, the Pythagorean identity plays a crucial role in simplifying to the final desired expression \(\cos^2 x - \sin^2 y\). Using the identity \(\cos^2 a = 1 - \sin^2 a\) and \(\sin^2 a = 1 - \cos^2 a\), we can express either trigonometric function in terms of the other. This substitution simplifies complex trigonometric calculations and transforms one kind of trigonometric function into another.

In step 3 of the solution, these identities are used to replace \(\cos^2 x\) and \(\sin^2 y\) parts of the expression obtained after simplification, verifying that the left side matches the right side of the original identity. Understanding how Pythagorean identities fit into trigonometric simplifications and transformations is key to mastering similar trigonometric problems.