Problem 22 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified using sum-to-product identities and simplification steps.

Step 1: Use Trigonometric Identities for Simplification

We begin by expressing both sides of the equation using trigonometric identities. For the left-hand side, which is \( \frac{\cos u - \cos v}{\cos u + \cos v} \), we use the sum-to-product identities:\[\cos u - \cos v = -2 \sin \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right) \\cos u + \cos v = 2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right) \]This allows us to rewrite the left-hand side as:\[\frac{-2 \sin \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)}{2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)}\]

Step 2: Simplify the Expression

Cancel common terms on the numerator and the denominator of the fraction:\[= -\frac{\sin \left(\frac{u+v}{2}\right)}{\cos \left(\frac{u+v}{2}\right)} \cdot \frac{\sin \left(\frac{u-v}{2}\right)}{\cos \left(\frac{u-v}{2}\right)}\]This simplifies to:\[= -\tan \left(\frac{u+v}{2}\right) \tan \left(\frac{u-v}{2}\right)\]This is the same expression as the right-hand side of the identity we need to verify.

Step 3: Verify the Identity

Both sides of the original equation now match:\[\frac{\cos u - \cos v}{\cos u + \cos v} = -\tan \left(\frac{u+v}{2}\right) \tan \left(\frac{u-v}{2}\right)\]Thus, we have verified that the identity holds true through trigonometric transformations and simplifications.

Key Concepts

Sum-to-Product Identities

The concept of sum-to-product identities plays a crucial role in trigonometry. These identities are particularly useful for transforming the sum or difference of trigonometric functions into products of functions. This can simplify complex expressions and make them easier to solve. For example, if you have \( \cos u - \cos v \), sum-to-product identities allow you to transform this into something more manageable:

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

These transformations can simplify a difficult identity verification or integration by converting sums into products. This was precisely how the left-hand side of the original identity was restructured using these identities.

Trigonometric Transformations

Trigonometric transformations are techniques used in solving trigonometric expressions. They involve changing the form without affecting the equality. These transformations often leverage identities to simplify or rewrite expressions to match other, more familiar forms.
In our original problem, the transformation was performed by using the sum-to-product identities. Initially, we had complex expressions with a sum involving cosines. By transforming it, we get:

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

This step makes it easier to identify further simplifications. By canceling identical terms from numerator and denominator, we transformed the expression to a product of tangents. Understanding such transformations can be a significant boost in verifying identities and solving trigonometric equations.

Verifying Trigonometric Identities

Verifying trigonometric identities is about showing that two sides of an equation are equivalent using trigonometric laws and rules. It often involves manipulation and transformation of one side to make it appear like the other.
In our exercise, once we transformed and simplified the original expression, we were able to match both sides:

• The left side became \( -\tan \left( \frac{u+v}{2} \right) \tan \left( \frac{u-v}{2} \right) \)
• The right side was already presented in that form

Verifying requires recognizing when two expressions are mathematically equivalent, even if they appear different at first sight. It’s like solving a puzzle where you rearrange pieces until both sides match perfectly. Understanding and mastering this skill requires practice in applying different identities and transformations.