Problem 90 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \tan y=\frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)} $$

Step-by-Step Solution

Answer

The identity is proven using sum-to-product identities.

Step 1: Understand the Given Problem

We need to prove the identity \( \tan y = \frac{\sin (x+y) - \sin (x-y)}{\cos (x+y) + \cos (x-y)} \). This means showing that both sides of the equation are equal by using trigonometric identities.

Step 2: Use Sum-to-Product Identities

Identify and apply the sum-to-product identities to simplify both the sine and cosine differences. The identities are:\[ \sin(a) - \sin(b) = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right) \]\[ \cos(a) + \cos(b) = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]Apply these identities to the expression: \( \sin(x+y) - \sin(x-y) = 2 \cos(x) \sin(y) \) \( \cos(x+y) + \cos(x-y) = 2 \cos(x) \cos(y) \).

Step 3: Simplify the Expression

Substitute the simplified forms into the original identity:\[ \frac{\sin(x+y) - \sin(x-y)}{\cos(x+y) + \cos(x-y)} = \frac{2 \cos(x) \sin(y)}{2 \cos(x) \cos(y)} \]The 2s and \( \cos(x) \) in the numerator and denominator cancel out, leaving:\[ \frac{\sin(y)}{\cos(y)} \]

Step 4: Recognize the Result

The resulting expression \( \frac{\sin(y)}{\cos(y)} \) is equivalent to \( \tan(y) \). This proves the identity as the left side of the equation is now equal to the right side.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \), is a fundamental trigonometric function. It is defined as the ratio of the sine to the cosine of an angle, specifically \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This relationship is crucial in understanding many trigonometric properties and identities, as it bridges two other critical trigonometric functions — sine and cosine.

• The tangent function is periodic, with a period of \( \pi \).
• It is undefined where the cosine function equals zero, leading to vertical asymptotes in its graph.
• The function itself can take any real number as a value, depending on the angle.

The tangent function is widely used in different fields, including engineering, physics, and computer science, for calculations involving right triangles and wave analysis. Understanding \( \tan \) and its properties helps in solving various trigonometric identities and equations.

Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric identities that convert sums into products. These identities are incredibly useful when simplifying expressions and solving trigonometric equations.
They can transform sine and cosine expressions involving sums or differences into products of trigonometric functions.

• For sine, the identity is \( \sin(a) - \sin(b) = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right) \).
• For cosine, the identity is \( \cos(a) + \cos(b) = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \).

When used effectively, these identities can simplify complex trigonometric expressions immensely.In the original solution, applying these identities directly transforms the terms \(\sin(x+y) - \sin(x-y)\) and \(\cos(x+y) + \cos(x-y)\) into more manageable forms. This leads directly to the simplification of the given equation to a familiar trigonometric expression.

Trigonometric Simplification

Trigonometric simplification is the process of making a trigonometric expression more straightforward or more easily manageable. This involves using various trigonometric identities and properties to rewrite expressions in simpler forms.
In the provided exercise, simplification through the use of identities is a critical step.

• By applying sum-to-product identities, complex trigonometric expressions are broken down into easier, equivalent forms.
• Canceling common terms, like the factor of \(2 \cos(x)\) in both the numerator and denominator, simplifies the fraction further.

The final product of this simplification in the exercise is the transformation of the expression \( \frac{2 \cos(x) \sin(y)}{2 \cos(x) \cos(y)} \) into \( \frac{\sin(y)}{\cos(y)} \).
This is recognized as \( \tan(y) \), which verifies the original identity, completing the proof. Such techniques not only help in solving equations but also deepen the understanding of the underlying concepts in trigonometry.