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The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \(x\), the sum of the squares of sine and cosine is equal to one:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity comes from the Pythagorean theorem in a right-angled triangle, where the legs are represented by sine and cosine values, and the hypotenuse is one unit long in the unit circle. Using this identity helps simplify trigonometric expressions.
In our exercise, it was used to simplify the expansion \( \sin^2 x + \cos^2 x + 2\sin x \cos x \). By recognizing \( \sin^2 x + \cos^2 x \) can be replaced with 1, we move closer to matching the identity with new forms, like \( 1 + \sin 2x \). This simplification is as straightforward as changing parts of an expression to a simpler form without losing equality. Using such identities helps in deriving different forms of the initial problem given in mathematical exercises.

The Double Angle Identity is a handy shortcut in trigonometry that transforms products of sine and cosine into a single trigonometric function. Specifically, for sine, the double angle identity is:

• \( \sin 2x = 2 \sin x \cos x \)

This shows that when the angle is doubled, its trigonometric sine representation can be expressed as a simple product of sine and cosine at the original angle. This is particularly useful in mathematical simplification efforts.When applied to our exercise, the expression \( 2 \sin x \cos x \) is recognized as \( \sin 2x \). As a result, we successfully rewrite \( 1 + 2 \sin x \cos x \) into \( 1 + \sin 2x \). This form reveals a clearer and more concise relationship that's equivalent to our prior expression in the given exercise. Such identities contribute significantly to simplifying and solving trigonometric forms.

Absolute value is a mathematical concept used to describe the magnitude of a number without considering its sign. In mathematical notation, it's represented by vertical bars as \( |x| \), which means:

• If \( x \geq 0 \), then \( |x| = x \)
• If \( x < 0 \), then \( |x| = -x \)

In essence, the absolute value of a number is its distance from zero on the number line, which is always non-negative.In trigonometry, when an expression such as \( \sin x + \cos x \) is put under a square root, we naturally consider its non-negative value, as square roots inherently result in non-negative outcomes. Thus, \( \sqrt{(\sin x + \cos x)^2} \) becomes \( |\sin x + \cos x| \). This ensures any negative value moods from the sine and cosine result are rendered positive, aligning neatly with square root behaviors.

A mathematical proof is a logical argument that demonstrates the truth of a given statement based on previously established axioms and theorems. The aim of a proof is to provide a clear path that shows why a certain equation or theorem holds valid. To structure a proof, consider these steps:

• Identify the known information and what needs to be proved.
• Use existing identities or theorems that might simplify the problem, like the Pythagorean and Double Angle Identities in our exercise.
• Manipulate the expressions diligently, always keeping track of the transformations used.
• Arrive at the statement needing proof in its simplest, most direct form.

In our exercise, the methodical steps taken used these principles: we began by expanding and simplifying expressions with known identities, eventually arriving at equivalent expressions. Each transformation adhered to logic that was grounded in existing mathematical laws. Ensuring clarity and correctness in each step of the proof is key to validating the final outcome.