Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \frac{1+\sin 2 v+\cos 2 v}{1+\sin 2 v-\cos 2 v}=\cot v $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \cot v \).

Step 1: Recall Trigonometric Identities

First, let's recall the relevant trigonometric identities. The double-angle formulas are: \( \sin 2v = 2 \sin v \cos v \) and \( \cos 2v = \cos^2 v - \sin^2 v \). Also, recall that \( \cot v = \frac{\cos v}{\sin v} \).

Step 2: Substitute Double-Angle Formulas

We substitute the double-angle formulas into the given identity. This gives us: \( \frac{1 + 2\sin v \cos v + (\cos^2 v - \sin^2 v)}{1 + 2\sin v \cos v - (\cos^2 v - \sin^2 v)} \).

Step 3: Simplify the Numerator

Let's simplify the numerator: \( 1 + 2\sin v \cos v + \cos^2 v - \sin^2 v = 1 + \cos^2 v + 2\sin v \cos v - \sin^2 v \). Group terms to become: \( (1 + \cos^2 v - \sin^2 v) + 2\sin v \cos v \).

Step 4: Simplify the Denominator

Simplify the denominator similarly: \( 1 + 2\sin v \cos v - \cos^2 v + \sin^2 v = 1 + \sin^2 v - \cos^2 v + 2\sin v \cos v \). Group terms to get: \( (1 + \sin^2 v - \cos^2 v) + 2\sin v \cos v \).

Step 5: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 v + \cos^2 v = 1 \). Substitute this to simplify further: \( 1 + \cos^2 v - \sin^2 v = 2\cos^2 v \) and \( 1 + \sin^2 v - \cos^2 v = 2\sin^2 v \).

Step 6: Factor and Simplify

Rewrite the fraction: \( \frac{2\cos^2 v + 2\sin v \cos v}{2\sin^2 v + 2\sin v \cos v} \). Factor out the common terms in both the numerator and denominator: \( \frac{2\cos v (\cos v + \sin v)}{2\sin v (\sin v + \cos v)} \). Cancel out the common \((\cos v + \sin v)\) terms: \( \frac{\cos v}{\sin v} \).

Step 7: Verify the Result

After canceling, the expression simplifies to \( \cot v \), thus proving the identity.

Key Concepts

Double-Angle Formulas

The double-angle formulas are essential tools in trigonometry. They allow us to express trigonometric functions of doubled angles in terms of functions of single angles. This transformation is especially useful for simplifying complex expressions and verifying identities.

### Key Double-Angle Formulas
Let's break down the most common double-angle formulas you'll need to know:

• For the sine function, the double-angle formula is: \[ \sin 2v = 2 \sin v \cos v \] This shows that the sine of a double angle is twice the product of the sine and cosine of the original angle.


• For the cosine function, two common forms are:
  \[ \cos 2v = \cos^2 v - \sin^2 v \]
  \[ \cos 2v = 2\cos^2 v - 1 \]
  \[ \cos 2v = 1 - 2\sin^2 v \] These forms are derived using the Pythagorean identity and are used based on convenience in calculations.

Understanding and using these formulas helps simplify trigonometric expressions, making it easier to verify identities or solve equations.

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It expresses a fundamental relationship between the sine and cosine functions.

### The Basic Identity
The most familiar form of the Pythagorean identity is:

• \[ \sin^2 v + \cos^2 v = 1 \] This indicates that, for any angle \(v\), the square sum of its sine and cosine equals one. This relation is derived from the Pythagorean theorem applied within the unit circle.

The identity is invaluable because it allows us to solve trigonometric equations by expressing sine in terms of cosine or vice versa. For instance, in the context of verifying trigonometric identities, you can substitute \(\sin^2 v\) or \(\cos^2 v\) using:

• \[ \sin^2 v = 1 - \cos^2 v \]
• \[ \cos^2 v = 1 - \sin^2 v \]

Using these substitutions can significantly simplify complex expressions and help prove or verify trigonometric identities, as seen in the problem we solved.

Cotangent Function

The cotangent function is frequently encountered in trigonometry and is the reciprocal of the tangent function. It's defined as the ratio of the cosine to the sine of an angle.

### Definition and Basic Properties

• Mathematically, the cotangent of an angle \(v\) is expressed as: \[ \cot v = \frac{\cos v}{\sin v} \]
• This means that the cotangent function is undefined whenever \(\sin v = 0\), as division by zero is not possible.

### Application in Verifying Identities

• In verifying identities, like the one addressed in the exercise, recognizing parts of the expression as a form of \(\cot v\) allows simplification.
• As seen in our solution, once the expression is simplified to \(\frac{\cos v}{\sin v}\), it is immediately recognized as \(\cot v\), hence proving the identity.

Understanding the relationships and properties of cotangent in conjunction with other trigonometric functions is invaluable when dealing with complex identities and equations.