Problem 34 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{\tan u-\tan v}{1+\tan u \tan v}=\frac{\cot v-\cot u}{\cot u \cot v+1}$$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\tan(u-v)}{1} = \frac{1}{\cot(u-v)} \).

Step 1: Understand the Identity to be Verified

We need to verify if the given identity \( \frac{\tan u - \tan v}{1 + \tan u \tan v} = \frac{\cot v - \cot u}{\cot u \cot v + 1} \) holds true. Our goal is to manipulate both sides to see if they are indeed equal.

Step 2: Work on the Left Side of the Identity

Start with the left side, \( \frac{\tan u - \tan v}{1 + \tan u \tan v} \). This can be identified as the tangent of the difference formula: \( \tan(u-v) = \frac{\tan u - \tan v}{1 + \tan u \tan v} \). Thus, the left side simplifies to \( \tan(u-v) \).

Step 3: Work on the Right Side of the Identity

The right side \( \frac{\cot v - \cot u}{\cot u \cot v + 1} \) can be identified as the co-function identity for cotangent, where \( \cot(u-v) = \frac{\cot v - \cot u}{\cot u \cot v + 1} \). This implies the right side is equivalent to \( \cot(u-v) \).

Step 4: Use the Pythagorean Identity

Recall the identity \( \tan x = \frac{1}{\cot x} \) and \( \cot x = \frac{1}{\tan x} \). When \( \tan(u-v) = 1/\cot(u-v) \), the two expressions are reciprocals, which implies their product is 1 given they simplify from different trigonometric identities.

Step 5: Conclusion of the Identity Verification

Since \( \tan(u-v) \) and \( \cot(u-v) \) are reciprocals for \( u eq v \), both produce the same numerical relationships for tangents and cotangents of angles, confirming they can be used to verify each other through the established identity transformation.

Key Concepts

Tangent Difference Formula

The tangent difference formula is a key trigonometric identity that simplifies the tangent of the difference between two angles. It is given by:

• \( \tan(u-v) = \frac{\tan u - \tan v}{1 + \tan u \tan v} \)

Understanding this formula helps break complex trigonometric expressions into simpler terms. The formula uses the tangents of individual angles \(u\) and \(v\), aligning with sine and cosine relationships.
The difference formula is particularly useful in verifying identities or solving equations involving tangent functions. It shows how changes in one angle affect the overall tangent. Visualizing the formula using small angle differences can also help build an intuitive grasp.
In practice, applying the tangent difference formula requires careful substitution and simplification. This highlights the importance of being familiar with core trigonometric identities, aiding in both simplifying expressions and understanding their geometric meaning.

Cotangent Formula

The cotangent formula brings to light the properties of the cotangent function, which is the reciprocal of the tangent function. The formula for the cotangent difference is:

• \( \cot(u-v) = \frac{\cot v - \cot u}{\cot u \cot v + 1} \)

This identity is essential in understanding how cotangents relate to one another when we subtract angles. Unlike tangent, cotangent is less frequently in standard equations, but it holds just as much significance in trigonometry.
The expression \( \frac{\cot v - \cot u}{\cot u \cot v + 1} \) simplifies the process of dealing with cotangent differences by using known values of \(u\) and \(v\).
Utilizing the cotangent formula in equations sheds light on the symmetry and periodicity in angle subtraction. It often allows swapping of angles or helps in deducing unknowns when working with fewer cotangent values.

Reciprocal Identities

Reciprocal identities are foundational in trigonometry, highlighting the relationship between trigonometric functions and their reciprocals:

• \( \tan x = \frac{1}{\cot x} \)
• \( \cot x = \frac{1}{\tan x} \)

These identities emphasize how the tangent and cotangent functions are interconnected. In solving equations, knowing that \( \tan x = \frac{1}{\cot x} \) helps solve for one when given the other.
The concept that these functions are reciprocals simplifies complex trigonometric problems by allowing conversion between tangent and cotangent. This exchange is particularly evident when dealing with various trigonometric transformations or simplifications.
Reciprocal identities contribute to the unity and symmetry in trigonometry, providing alternate pathways to verify identities or solve equations that could otherwise become cumbersome.

Pythagorean Identity

The Pythagorean identities are indispensable tools in trigonometry, providing a relationship between tangent and cotangent:

• One essential identity is \( \tan^2 x + 1 = \sec^2 x \)
• Another involves cotangent: \( 1 + \cot^2 x = \csc^2 x \)

These identities are rooted in the Pythagorean theorem, focusing on the inherent relationships between angles and sides in right-angled triangles. They are crucial in simplifying expressions such as \( \tan(u-v) \) and \( \cot(u-v) \).
Understanding Pythagorean identities helps solve trigonometric equations and verify identities. They act as bridges between trigonometric functions representing different geometric and algebraic forms.
This identity particularly resolves conflicts in verifying if two expressions, such as those involving tangent and cotangent, are equal, as it often helps transition between different trigonometric forms.