Problem 29 - Precalculus Essentials

Question

In Exercises \(23-34\), verify each identity. $$\cot x=\frac{\sin 2 x}{1-\cos 2 x}$$

Step-by-Step Solution

Answer

The given trigonometric equation \( \cot x = \frac{\sin 2x}{1-\cos 2x} \) is a valid identity.

Step 1 - Rewrite the Right Side

Rewrite the right side of the equation using the double angle identities, \( \sin 2x = 2\sin x \cos x \) and \( \cos 2x = 1 - 2\sin^2x \). Thus the right side becomes \( \frac{2\sin x \cos x}{1 - (1 - 2\sin^2x)} = \frac{2\sin x\cos x}{2\sin^2x} \).

Step 2 - Simplify the Expression

Simplify the expression obtained in the previous step by cancelling out the common factors. You will obtain \( \cot x = \frac{\cos x}{\sin x} \).

Step 3 - Recognize the Left Side

Now notice that \( \frac{\cos x}{\sin x} \) is exactly the definition of \( \cot x \) which is the left side of our initial equation. Thus, we have verified that \( \cot x = \frac{\sin 2x}{1-\cos 2x} \).

Key Concepts

Cotangent Identity

The cotangent identity is a trigonometric identity that expresses the relationship between sine and cosine through cotangent. Cotangent is essentially the reciprocal of tangent and is defined as:

• \( \cot x = \frac{1}{\tan x} \)
• Which leads to: \( \cot x = \frac{\cos x}{\sin x} \)

By clearly understanding that \( \cot x \) is the ratio of the cosine to the sine of an angle, we can manipulate and derive various other identities.
For instance, in the expression \( \cot x = \frac{\sin 2x}{1 - \cos 2x} \), rewriting it based on trigonometric identities allows us to reach familiar forms such as \( \frac{\cos x}{\sin x} \).
Understanding this basic definition and the transformations between different forms of trigonometric identities helps in simplifying complex trigonometric equations.

Double Angle Identities

Double angle identities are equations that involve trigonometric functions at twice a given angle. These identities help simplify complex trigonometric expressions and are essential in various mathematical calculations.
Some key double angle identities include:

• \( \sin 2x = 2 \sin x \cos x \)
• \( \cos 2x = \cos^2 x - \sin^2 x \)
• Alternative forms of \( \cos 2x \) are: \( 1 - 2\sin^2 x \) and \( 2\cos^2 x - 1 \)

These forms allow you to use the identity that best suits your problem.
In the exercise, we used \( \sin 2x = 2 \sin x \cos x \) and the alternative form \( \cos 2x = 1 - 2\sin^2 x \) to simplify and verify the given trigonometric identity.
The application of double angle identities provides an elegant way to handle trigonometric expressions, resolving them into more manageable forms.

Verifying Identities

Verifying trigonometric identities involves proving that two sides of an equation represent the same value for all values of the variable in their domains. This process often requires recognizing, transforming, and simplifying expressions to show equivalence.
In the task of verifying \( \cot x = \frac{\sin 2x}{1 - \cos 2x} \), we aim to transform the expression on one side into the form of the other. Here's how verifying identities typically works:

• Start by rewriting one side of the equation using known identities.
• Simplify the expression step by step, applying algebraic manipulations and trigonometric identities, such as double angle identities.
• Continue simplifying until both sides of the equation match.

Verifying identities strengthens your understanding of trigonometric relationships and enhances your problem-solving skills by encouraging you to see alternate forms of the same function. It is a practice that not only verifies mathematical equivalence but deepens comprehension of how trigonometric functions are interrelated.