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Understanding the sine sum formula is essential for solving a wide range of trigonometry problems. This formula is used to expand expressions like \(\sin(A + B)\), where A and B are angles. The sine sum formula states that \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\). This formula is powerful because it allows you to break down complex trigonometric expressions into simpler terms that are easier to evaluate.

In the example provided, we apply this formula to the expression \(\sin \left(x+\frac{3 \pi}{2}\right)\). By substituting A with x and B with \(\frac{3 \pi}{2}\), we expand it as \(\sin(x)\cos\left(\frac{3 \pi}{2}\right) + \cos(x)\sin\left(\frac{3 \pi}{2}\right)\). This step is critical as it sets the stage for simplification and helps in verifying the given trigonometric identity.

Trigonometric functions like sine and cosine are fundamental in mathematics and define relationships between the angles and sides of triangles. They also extend to the unit circle, which allows us to evaluate trigonometric functions for a broad range of angles beyond the initial 0 to 90 degrees.

Especially important are the specific values of these functions at angles such as \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\). For instance, as seen in our example, \(\cos\left(\frac{3\pi}{2}\right)\) equals 0 and \(\sin\left(\frac{3\pi}{2}\right)\) equals -1. These values play a key role in simplifying trigonometric expressions and are often memorized as they frequently arise in various mathematical contexts.

Identity verification in trigonometry requires you to apply known identities and trigonometric properties to prove whether a given statement is true for all values within the domain of the variables involved. In the exercise example, to verify the identity \(\sin \left(x+\frac{3 \pi}{2}\right) = -\cos(x)\), we first expanded the left side using the sine sum formula. The next step was to evaluate trigonometric functions at known angle values, which simplified our expression.

Finally, the simplified form was compared to the right side of the given identity to ensure they match. If both sides of the equation are identical after simplification, the identity is successfully verified. This practice strengthens understanding of trigonometric functions and their interrelations, which is a valuable skill in algebra, calculus, and beyond.