Problem 20 - Precalculus: Functions and Graphs

Question

Verify the identity. $$\frac{\sin u+\sin v}{\cos u+\cos v}=\tan \frac{1}{2}(u+v)$$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Understanding the Task

We are given an equation \( \frac{\sin u + \sin v}{\cos u + \cos v} = \tan \frac{1}{2}(u+v) \) and need to verify if it is an identity. To verify it, we'll try to simplify the left-hand side to match the right-hand side.

Step 2: Using Sum-to-Product Identities

Recall the sum-to-product identities: \( \sin a + \sin b = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right) \) and \( \cos a + \cos b = 2 \cos \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right) \). Apply these identities to simplify the numerator and the denominator.

Step 3: Simplifying the Numerator

For the numerator, use \( \sin u + \sin v = 2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right) \). This breaks down the complex sum into a product of sines and cosines.

Step 4: Simplifying the Denominator

For the denominator, use \( \cos u + \cos v = 2 \cos \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right) \). Similar to the numerator, it reduces to a product.

Step 5: Substituting the Simplifications

Substitute the simplified forms back into the expression: \( \frac{2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)}{2 \cos \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)} \). Notice now that the \(2\) and \( \cos \left( \frac{u-v}{2} \right) \) terms cancel each other out.

Step 6: Result After Cancellation

After canceling out, we have \( \frac{\sin \left( \frac{u+v}{2} \right)}{\cos \left( \frac{u+v}{2} \right)} \), which simplifies to \( \tan \left( \frac{u+v}{2} \right) \). This matches the right-hand side.

Key Concepts

Sum-to-Product Identities

Sum-to-Product identities are useful trigonometric identities that convert sums of sines and cosines into products. This can often make complex trigonometric expressions easier to work with. For instance, we have the identities:

• \( \sin a + \sin b = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right) \)
• \( \cos a + \cos b = 2 \cos \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right) \)

These identities are incredibly handy when you have expressions involving sums in the numerator or the denominator.
They allow you to express these sums as products, which could simplify fractional expressions.
In the exercise, sum-to-product identities help simplify both the numerator (\( \sin u + \sin v \)) and the denominator (\( \cos u + \cos v \)).
This simplification is a crucial step in verifying that the initial trigonometric equation is an identity.

Angle Addition Formulas

Angle addition formulas allow you to express the sine or cosine of the sum (or difference) of two angles in terms of the sines and cosines of those angles separately.

• For sine, the formula is: \( \sin(a+b) = \sin a \cos b + \cos a \sin b \)
• For cosine, it is: \( \cos(a+b) = \cos a \cos b - \sin a \sin b \)

In this exercise, while we are not directly using the angle addition formulas, they play an underpinning role in understanding how the sum-to-product identities are derived.
Angles are a foundational element of understanding trigonometric expressions and the way these formulas manipulate them.The addition or subtraction of angles is a regular operation in trigonometry that often leads to fruitful simplifications, just like how it allows breaking down and constructing meaningful expressions in algebraic methods.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions into simpler forms.
It aims at making these expressions easier to work with, especially in equations and identities.
In the context of our problem, this involves several interconnected steps:

• Simplifying expressions using identities, such as the sum-to-product identities.
• Canceling common terms like the multiplicative terms in the numerator and denominator.
• Recognizing patterns that match known trigonometric identities, like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

When dealing with identities, like the one in your exercise, trigonometric simplification ensures that both sides of the equation are identical.
It transforms a complex mathematical expression into a simpler form to demonstrate that one side can indeed be expressed as the other, demonstrating the true identity nature of the equation.