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The tangent identity is a fundamental concept in trigonometry and is essential in solving many problems involving angles. It can be expressed as \[\tan x = \frac{\sin x}{\cos x}\]This means that the tangent of an angle \(x\) in a right triangle is the ratio of the sine of that angle over the cosine of the same angle. Understanding this fundamental identity is crucial because it connects the tangent function with the sine and cosine functions, making it easier to manipulate and solve trigonometric equations.

When verifying an identity like the given problem, understanding how to apply the tangent identity allows you to transform complex expressions into simpler forms, making problem-solving more manageable. Here, for instance, we use the tangent identity to express the right-hand side of the equation in terms of sine and cosine. This simplification builds the foundation for comparing both sides of the equation throughout the rest of the solution.

The sine of the sum identity describes how to find the sine of the sum of two angles and is important in trigonometry for solving addition or subtraction problems involving angles. The identity is written as:\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]This means that the sine of the sum of two angles is equal to the sum of their sines and cosines in this specific arrangement. By splitting a sum into individual angles, this identity helps you to analyze and simplify trigonometric expressions.

When applied in the context of the original exercise, understanding and utilizing the sine of sum identity helps break down compound angles, enabling us to express them as simpler expressions. By adjusting the terms in the numerator of our target expression using this identity, we connect these transformations, translating complicated expressions into a more familiar format which is invaluable for the comparison and verification against the original equation.

The cosine of sum identity is an equally important trigonometric identity that assists in expressing the cosine of a sum of two angles. It is written as:\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]This formula shows that the cosine of the sum of two angles can be decomposed into the product of their individual cosines minus the product of their sines. It plays a crucial role in simplifying and solving equations and transformations in trigonometry.

In the given problem, the cosine of sum identity is utilized to handle the denominator of the target expression. By using this identity, we can rewrite compound angles into components that resemble known trigonometric expressions, helping us to adjust for a similar form to facilitate comparison and equality checks. These identities act as key tools for converting complex expressions into simpler forms, allowing deeper insight and aiding in solving trigonometric problems efficiently.