Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\frac{\sin (A-B)}{\sin B}+\frac{\cos (A-B)}{\cos B}=\frac{\sin A}{\sin B \cos B}$$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to the same expression.

Step 1: Simplify the Left Side of the Equation

The left side of the equation is \( \frac{\sin(A-B)}{\sin B} + \frac{\cos(A-B)}{\cos B} \). We can use the angle subtraction formulas: \(\sin(A-B) = \sin A \cos B - \cos A \sin B\) and \(\cos(A-B) = \cos A \cos B + \sin A \sin B\). Replace these identities back in the equation.

Step 2: Substitute Angle Subtraction Formulas

Substitute the angle subtraction formulas into the equation:\[ \frac{\sin A \cos B - \cos A \sin B}{\sin B} + \frac{\cos A \cos B + \sin A \sin B}{\cos B} \]

Step 3: Separate Fractions

Separate the terms in the fractions:\[ \frac{\sin A \cos B}{\sin B} - \frac{\cos A \sin B}{\sin B} + \frac{\cos A \cos B}{\cos B} + \frac{\sin A \sin B}{\cos B} \].

Step 4: Simplify Separate Fractions

Simplify each fraction: \(\frac{\sin A \cos B}{\sin B} = \sin A \frac{\cos B}{\sin B}\), \(\frac{\cos A \sin B}{\sin B} = \cos A\), \(\frac{\cos A \cos B}{\cos B} = \cos A\), and \(\frac{\sin A \sin B}{\cos B} = \sin A \frac{\sin B}{\cos B}\).

Step 5: Combine Like Terms

Combine and simplify like terms: \[ \sin A \frac{\cos B}{\sin B} + \sin A \frac{\sin B}{\cos B} \]. Notice that this simplifies to \(\frac{\sin A}{\sin B\cos B}\) after factoring \(\sin A\).

Step 6: Verify Result

After simplification, the left side of the original equation \(\frac{\sin(A-B)}{\sin B} + \frac{\cos(A-B)}{\cos B}\) simplifies to \(\frac{\sin A}{\sin B \cos B}\), which matches the right side of the given equation.

Key Concepts

Angle Subtraction Formulas

When dealing with trigonometric identities, angle subtraction formulas play an essential role. These formulas help us express trigonometric functions of subtracted angles in simpler terms. The angle subtraction formulas are:

• For sine: \( \sin(A-B) = \sin A \cos B - \cos A \sin B \)
• For cosine: \( \cos(A-B) = \cos A \cos B + \sin A \sin B \)

These formulas allow us to break down complex expressions involving angles into more manageable parts.
By substituting these formulas into an expression, like in the given exercise, we enable further simplification and verification of identities. Understanding these formulas is crucial because they form the essence of transforming and simplifying various trigonometric equations.

Fraction Simplification

In mathematics, simplifying fractions is often a necessary step to make equations easier to solve. In trigonometric equations, this involves expressing the fractions in their simplest form. Let's consider the process of simplifying a trigonometric fraction.
In Step 3 of the solution, we took the trigonometric terms and divided them into separate fractions. For instance, splitting the fraction \( \frac{\sin A \cos B}{\sin B} \) allowed us to rewrite it as \( \sin A \times \frac{\cos B}{\sin B} \). This step helps isolate the trigonometric functions, setting the stage for further simplification.

• Transform complex fractions into simpler terms by reducing terms where possible.
• Combine like terms such as \( \cos A \) and \( \cos A \) while keeping track of signs and coefficients.

Mastering these techniques is essential for verifying complex trigonometric identities, as shown in the exercise.

Verifying Identities

Verifying trigonometric identities is like solving a puzzle. The goal is to show that both sides of an equation are equivalent. In the given exercise, the problem asks us to verify the identity:
\[\frac{\sin (A-B)}{\sin B} + \frac{\cos (A-B)}{\cos B} = \frac{\sin A}{\sin B \cos B}\]This requires simplifying the left side until it matches the right side of the equation.

• Apply known formulas like angle subtraction to express complex terms in simpler forms.
• Use algebraic manipulation such as simplifying fractions and combining like terms to transform the expression.
• Check each step carefully, ensuring that each transformation stays true to the original identity.

Once all steps are correctly followed, you'll end up with the same expression on both sides, confirming the identity's validity. This step-by-step process is key to building a strong foundation in trigonometry.