Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\cot 4 u-1}{\cot 4 u+1}=\frac{1-\tan 4 u}{1+\tan 4 u} $$

Step-by-Step Solution

Answer

The identity is verified as true.

Step 1: Understand the Identity

We need to prove that \( \frac{\cot 4u - 1}{\cot 4u + 1} = \frac{1 - \tan 4u}{1 + \tan 4u} \). This involves showing both sides of the equation are equal using some algebraic manipulations and trigonometric identities.

Step 2: Use Cotangent and Tangent Relationship

Recall the identity \( \cot x = \frac{1}{\tan x} \). We can apply this identity for \( \cot 4u \) and express it in terms of \( \tan 4u \) as \( \cot 4u = \frac{1}{\tan 4u} \).

Step 3: Substitute Cotangent Into the Left Side

Substitute \( \cot 4u = \frac{1}{\tan 4u} \) into the left side of the equation: \( \frac{\cot 4u - 1}{\cot 4u + 1} = \frac{\frac{1}{\tan 4u} - 1}{\frac{1}{\tan 4u} + 1} \).

Step 4: Simplify the Fraction

Simplify the fraction by multiplying the numerator and the denominator by \( \tan 4u \), which gives us: \( \frac{1 - \tan 4u}{1 + \tan 4u} \). This transformation shows that the left-hand side has been successfully converted to the right-hand side.

Step 5: Conclude the Verification

Since both sides of the original identity simplify to \( \frac{1 - \tan 4u}{1 + \tan 4u} \), the identity is verified as true.

Key Concepts

Understanding Cotangent

The cotangent, often denoted as \( \cot \theta \), is a fundamental trigonometric function that serves as the reciprocal of the tangent. This means that:

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

Understanding cotangent is crucial when simplifying trigonometric expressions as it allows for alternative presentations of the same value, particularly when comparing angles or identities.
In trigonometry, cotangent is beneficial for converting between trig functions, which is especially useful in proving identities like the one presented in our exercise.

For instance, substituting \( \cot 4u = \frac{1}{\tan 4u} \) allows one to harness the algebraic properties of fractions to simplify complex expressions or identities. This relationship between cotangent and tangent is often employed in exercises that require verification or transformation of identities.

The Tangent Function

The tangent function, symbolized as \( \tan \theta \), is one of the primary trigonometric functions, commonly defined in the context of a right triangle as the ratio of the opposite side to the adjacent side.
It's also expressed using sine and cosine functions as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

In the exercise, we're looking to verify an identity involving transformations of tangent expressions, which often require an understanding of how tangent operates within general identities.

The tangent function exhibits periodic properties, repeating every \( \pi \) radians, which can be crucial when considering multiple angles like \( 4u \), as seen in our given identity.
Transforming \( \tan 4u \) into \( 1 - \tan 4u \) or similar forms involves keeping track of these properties to accurately manipulate or simplify expressions.
The interchange between tangent and other trigonometric functions, including by using their reciprocals (like in the cotangent), often forms the backbone of both proving and simplifying trigonometric identities.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves using algebraic rules and procedures to transform or simplify mathematical expressions.
In the context of trigonometry, algebraic manipulation often includes:

• Substitution of equivalent forms (e.g., \( \cot x = \frac{1}{\tan x} \))
• Multiplying or dividing terms by common factors (as we did with \( \tan 4u \) in the solution)
• Rearranging equations or identities for simplification

In verifying our identity, crucial steps involved substituting \( \cot 4u \) with its tangent form and then algebraically simplifying the resulting expression.

Multiplying the fractions by \( \tan 4u \), we bypass terms that appeared initially complicated, showcasing how algebraic manipulation can streamline the verification process.
Through such methods, various trigonometric identities are either simplified or proven, with careful attention to maintaining equivalence throughout the steps.
Understanding the balance and symmetry in algebraic properties is key to successful manipulation, making those techniques indispensable in solving or proving trigonometric problems.