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Trigonometric identities are fundamental equalities that involve trigonometric functions such as sine, cosine, and tangent. These identities enable you to simplify expressions and solve trigonometric equations more easily. In mathematics, trigonometric identities are numerous, meaning there are various ways through which trigonometric expressions can be transformed or rewritten.

• Basic Identities: These include Pythagorean identities like \( \sin^2x + \cos^2x = 1 \).
• Reciprocal Identities: Such as \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \).
• Quotient Identities: \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

By applying these identities, you can resolve complex trigonometric problems. They are the building blocks for deriving more complex formulas like the double-angle and sum-to-product identities. Recognizing these identities within expressions allows you to manipulate and simplify them, as shown in the given exercise.

The difference of squares is a key algebraic identity that states \( a^2 - b^2 = (a-b)(a+b) \). This concept is crucial as it provides a method for breaking down certain polynomial expressions, making them easier to handle and simplify. The exercise exemplifies this.For the expression \((\sin x - \cos x)(\sin x + \cos x)\), you identify the structural form of a difference of squares:

• Set \( a = \sin x \) and \( b = \cos x \).
• Apply the formula to yield \( \sin^2x - \cos^2x \).

Utilizing this identity in trigonometry allows us not only to simplify but also to connect with other identities, such as those seen in the later steps of the double-angle formulas.

The cosine double-angle formula is an integral identity in trigonometry, expressed as \( \cos(2x) = \cos^2x - \sin^2x \). It provides a way to express trigonometric functions of double angles in terms of functions of single angles. In this exercise, the transformation of \( \sin^2x - \cos^2x \) to \( -\cos(2x) \) highlights the use of the cosine double-angle formula.Here's how it connects:

• The original products \( \sin^2x - \cos^2x \) are part of the double-angle identity transformations.
• This formula also has other equivalent forms: \( \cos(2x) = 1 - 2\sin^2x \) and \( \cos(2x) = 2\cos^2x - 1 \).
• The sign change (to negative) occurs due to the initial order of \( \sin^2x - \cos^2x \) compared to \( \cos^2x - \sin^2x \).

The double-angle formula for cosine allows for a deeper understanding and proves highly usable when dealing with trigonometric expressions involving angles that are twice another. It's an essential technique that can simplify many complex trigonometric problems.