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The Pythagorean identity is one of the most fundamental trigonometric identities. It states that for any angle \(x\), the expression \( \sin^2 x + \cos^2 x = 1 \) always holds true. This identity is rooted in the Pythagorean theorem, hence its name, and plays a critical role in simplifying trigonometric expressions.

When you're working on an equation like \((\sin x+ \cos x)^{2}=1+\sin 2 x\), the Pythagorean identity is crucial. In our example, expanding \((\sin x+ \cos x)^{2}\) gives terms \(\sin^2 x\) and \(\cos^2 x\). According to the Pythagorean identity, you can directly replace \(\sin^2 x + \cos^2 x\) with 1.

This drastically simplifies our work, reducing the expression further, making it easier to verify if the expressions on both sides are equal. Remember, spotting opportunities to apply identities like this can make solving trigonometric equations more straightforward.

LHS and RHS refer to the Left Hand Side and the Right-Hand Side of an equation, respectively. When solving or verifying an equation, such as \((\sin x+ \cos x)^{2}=1+\sin 2 x\), one common approach is to work separately on the LHS and RHS to simplify them.

In our task, we began by expanding the LHS: \((\sin x+ \cos x)^{2}\). Through expansion using the formula \((a+b)^2 = a^2 + 2ab + b^2\), we obtained \(\sin^{2}x + 2 \sin x \cos x + \cos^{2}x\). Next, simplifying using the Pythagorean identity, we arrived at \(1 + 2 \sin x \cos x\).

Then, converting the RHS \(1 + \sin 2x\) using the double angle formula \(\sin 2x = 2 \sin x \cos x\), we also found \(1 + 2 \sin x \cos x\). Since both sides simplify to the same expression, \(1 + 2 \sin x \cos x\), the identity is verified, confirming the equation is balanced.

Double angle formulas are powerful tools in trigonometry that express trigonometric functions of double angles (\(2x\)) in terms of single angles (\(x\)). One of the most common double angle formulas is for sine: \(\sin 2x = 2 \sin x \cos x\). This formula helps bridge expressions involving angles like \(2x\) with those involving angles like \(x\).

In the identity \((\sin x+ \cos x)^{2}=1+\sin 2 x\), applying the double angle formula is a crucial step. By converting \(\sin 2x\) on the RHS to \(2 \sin x \cos x\), we align both sides with similar terms involving \(\sin x\) and \(\cos x\).

This transformation is essential for showing equality between the altered LHS and RHS, making the verification process smoother. Understanding these formulas allows for simplification and problem-solving in many trigonometric contexts, opening the pathway to easier manipulation of complex expressions.