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The addition formula for sine is a key trigonometric identity that allows us to simplify expressions involving the sine of a sum. The formula is expressed as follows:

• \( \sin (r + s) = \sin (r) \cos (s) + \cos (r) \sin (s) \)

This identity helps break down a complex sine expression into simpler parts, which involve the sine and cosine of the individual angles \(r\) and \(s\). By applying this identity, you can transform expressions and make comparisons easier.

In the problem given, the left-hand side of the equation involves \( \sin (r+s) \). By applying the addition formula, it becomes possible to rewrite the expression in terms of \( \sin (r) \), \( \cos (s) \), \( \cos (r) \), and \( \sin (s) \). This restructuring is essential for verifying whether the given equation is an identity. It provides a method to map components from both sides of an equation in a structured way.

The subtraction formula for sine is another critical identity in trigonometry, enabling us to handle the sine of an angle difference. This identity is given by:

• \( \sin (r - s) = \sin (r) \cos (s) - \cos (r) \sin (s) \)

This formula is particularly useful in rewriting and simplifying trigonometric equations that involve subtractions to provide a form that is more analyzable.

In the particular exercise, we see \( \sin (r-s) \) in the equations provided. By using the subtraction formula, the equation can be split into separate terms involving \( \sin \) and \( \cos \) of the individual angles \(r\) and \(s\). Because of this breakdown, the equation can be manipulated to reflect and potentially verify whether it equates to an identity.

This step, much like the addition case, helps to realign expressions so they can be directly compared to the simplified forms on the right-hand side of the equation.

The tangent function is a fundamental part of trigonometry, often involved in many identities and geometric calculations. It is defined as:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

This ratio of the sine and cosine functions helps in expressing relationships between the sides of a right-angled triangle and the angles. The tangent function arises in various identities and can be used to transform expressions.

In the exercise, the goal was to verify expressions as identities by relating to the tangent function. Rewriting the original expressions involving sine into terms involving tangent was a crucial step. By simplifying \( \frac{\sin (r+s)}{\cos(r)\cos(s)} \) and \( \frac{\sin (r-s)}{\cos(r)\cos(s)} \), we discover the corresponding tangent expressions on the right-hand side (\( \tan(r) + \tan(s) \) and \( \tan(r) - \tan(s) \)).

This simplification not only aids in recognizing identities but also in understanding how expressions evolve in trigonometric forms, underlining the pivotal role that the tangent function plays in linking different trigonometric expressions.