Problem 21 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The trigonometric identity is verified as both sides simplify to the same expression.

Step 1: Understand the Identity

We need to verify the identity \( \frac{\sin u - \sin v}{\sin u + \sin v} = \frac{\tan \frac{1}{2}(u - v)}{\tan \frac{1}{2}(u + v)} \). It involves simplifying trigonometric expressions using well-known identities.

Step 2: Apply Sum-to-Product Identities

Recall the sum-to-product identities: \( \sin u - \sin v = 2 \cos \left( \frac{u+v}{2} \right) \sin \left( \frac{u-v}{2} \right) \) and \( \sin u + \sin v = 2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right) \). Substitute these into the left side of the identity.

Step 3: Substitute and Simplify the Left Side

Substitute the identities into the left ratio: \( \frac{2 \cos \left( \frac{u+v}{2} \right) \sin \left( \frac{u-v}{2} \right)}{2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)} \). Simplifying by canceling the '2' terms gives \( \frac{\cos \left( \frac{u+v}{2} \right) \sin \left( \frac{u-v}{2} \right)}{\sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)} \).

Step 4: Use the Tangent Half-Angle Identity

Recall the relation for tangent half-angle: \( \tan \frac{1}{2} x = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \). Apply this to both the numerator and denominator: the numerator becomes \( \tan \frac{1}{2}(u-v) \) and the denominator becomes \( \tan \frac{1}{2}(u+v) \).

Step 5: Verify the Identity

Now both sides of the original identity are: \( \frac{ \tan \frac{1}{2}(u-v) }{ \tan \frac{1}{2}(u+v) } \). Since these expressions are equal, the identity is verified.

Key Concepts

Sum-to-Product Identities

Sum-to-Product Identities are a set of trigonometric formulas used to transform sums or differences of sines and cosines into products. This is particularly useful in simplifying expressions and solving equations involving trigonometric functions.
They are derived from the addition and subtraction formulas for sine and cosine. For instance, the identity for the difference of two sine functions is given by:

• \( \sin u - \sin v = 2 \cos \left( \frac{u+v}{2} \right) \sin \left( \frac{u-v}{2} \right) \)

Likewise, the identity for the sum of two sine functions is:

• \( \sin u + \sin v = 2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right) \)

Using these identities helps in breaking down complex expressions into simpler forms, allowing us to easily manipulate and solve them.
When verifying identities, like in our exercise, these transformations become extremely helpful in matching expressions from different sides of an equation. Understanding how to appropriately apply these identities is key to mastering trigonometry.

Tangent Half-Angle Identity

The Tangent Half-Angle Identity is a useful tool in trigonometry that relates the tangent of half an angle to sine and cosine. This identity is particularly significant when verifying or simplifying expressions involving tangents.
The identity is stated as:

• \( \tan \frac{1}{2} x = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \)

This provides a way to express tangent entirely in terms of sine and cosine. This identity is instrumental when simplifying ratios of sines and cosines, as seen in this exercise.
When we use it to convert the trigonometric expressions, such as in the numerator and denominator of complex expressions, it helps verify identities by transforming the left and right sides of equations into comparable terms. Mastery of this identity allows one to manipulate trigonometric expressions efficiently, greatly simplifying problem-solving.

Trigonometric Simplification

Trigonometric Simplification involves reducing a complex trigonometric expression into a simpler form, often by using trigonometric identities. This process is crucial in verifying identities and solving trigonometric equations more easily.
Simplification often involves:

• Converting sums and differences into products using Sum-to-Product Identities.
• Using identities like the Tangent Half-Angle to switch between different trigonometric forms.
• Cancelling common terms in fractions to reduce clutter and focusing on core trigonometric forms.

For instance, in the original step-by-step solution, each term of the fraction was simplified using identities, and then common terms were canceled, leading to a much simpler expression.
This approach is a systematic way to handle trig ratios by breaking them into smaller, more manageable parts. Learning these simplification techniques can dramatically ease the understanding and solving of complex trigonometric problems.