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The Pythagorean Identity is a fundamental relation in trigonometry and is essential for various trigonometric proofs and calculations. It states that for any angle \( x \), the sum of the squares of the sine and cosine functions is always equal to one. Mathematically, it is expressed as: \[ \sin^2 x + \cos^2 x = 1 \] This identity arises from the Pythagorean Theorem applied to the unit circle, where the radius equals one. In the context of the problem, we used this identity to simplify the expanded expression \( \sin^2 x + 2\sin x \cos x + \cos^2 x \) by substituting \( \sin^2 x + \cos^2 x \) with 1. This simplification plays a crucial role in verifying the trigonometric identity by showing that the left-hand side expression equals the right-hand side.

The Distributive Property is one of the core properties of arithmetic that also holds true in algebra, including trigonometry. It states that multiplying a sum by a number gives the same result as multiplying each addend separately by the number and then adding the results. Mathematically, it can be shown as: \[ a(b + c) = ab + ac \] In the exercise, we employed the distributive property to expand \((\sin x + \cos x)^2\). We multiplied each part of the expression \((\sin x + \cos x)\) with itself to get:

• \(\sin^2 x \)
• \(2\sin x \cos x \)
• \(\cos^2 x\)

This expansion simplifies the process of verification by forming an expression that makes applying the Pythagorean Identity straightforward.

Verifying trigonometric identities involves proving that both sides of a given equation are equivalent through algebraic manipulation and the use of known identities. This process helps deepen understanding of trigonometric relationships and ensures accuracy in mathematical reasoning.
To verify a trigonometric identity, it's often helpful to:

• Simplify each side separately using algebraic techniques.
• Apply known trigonometric identities, like the Pythagorean Identity, to assist in simplification.
• Ensure that transformations maintain the equality of the left and right sides.

In this exercise, the identity \((\sin x + \cos x)^2 = 1 + 2\sin x \cos x\) was verified by expanding the left side and then simplifying using the Pythagorean Identity, demonstrating that both sides of the equation indeed match. Such exercises enhance one's ability to manipulate and comprehend the coherence of trigonometric expressions.