Problem 134 - Precalculus

Question

For the following exercises, prove the identity given. $$ \cot x-\tan x=2 \cot (2 x) $$

Step-by-Step Solution

Answer

The identity \( \cot x - \tan x = 2\cot(2x) \) is true.

Step 1: Express Cotangent and Tangent in terms of Sine and Cosine

The identities for cotangent and tangent in terms of sine and cosine are \( \cot x = \frac{\cos x}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Let's rewrite the left side of the equation using these identities: \[ \cot x - \tan x = \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}.\]

Step 2: Find a Common Denominator

To combine these fractions, we need a common denominator. The common denominator of \( \sin x \) and \( \cos x \) is \( \sin x \cos x \). Rewrite each fraction with this common denominator:\[ \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x \cos x} \]\[ \frac{\sin x}{\cos x} = \frac{\sin^2 x}{\sin x \cos x}. \]

Step 3: Subtract the Fractions

Now subtract the two fractions:\[ \frac{\cos^2 x}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x} = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}.\]

Step 4: Utilize Trigonometric Identities

Recall the identity \( \cos 2x = \cos^2 x - \sin^2 x \).Substitute this identity into the expression:\[ \frac{\cos^2 x - \sin^2 x}{\sin x \cos x} = \frac{\cos 2x}{\sin x \cos x}.\]

Step 5: Simplify the Expression

Notice that the right side of the original identity is \( 2\cot(2x) \). The identity for \( \cot(2x) \) is \( \frac{\cos 2x}{\sin 2x} \). Given \( \sin 2x = 2 \sin x \cos x \), we have:\[ \cot(2x) = \frac{\cos 2x}{2 \sin x \cos x}. \]Thus, \( 2\cot(2x) \) is:\[ 2 \times \frac{\cos 2x}{2 \sin x \cos x} = \frac{\cos 2x}{\sin x \cos x}. \]

Step 6: Confirm the Identity

Since the simplified left side \( \frac{\cos 2x}{\sin x \cos x} \) is identical to the expression we found for \( 2\cot(2x) \), this confirms that:\[ \cot x - \tan x = 2 \cot(2x). \]

Key Concepts

Cotangent

The cotangent function, often abbreviated as "cot," is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function. This means that instead of looking at the ratio of the opposite side to the adjacent side in a right triangle (as tangent does), cotangent is the ratio of the adjacent side to the opposite side. Mathematically, we express it as:

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

You can visualize cotangent as adjacent to opposite in a triangle or as the quotient of cosine to sine for an angle in the unit circle.This perspective helps in understanding its behavior and relationship with other trigonometric functions.Since cotangent is the reciprocal of tangent, it becomes undefined wherever the tangent is zero.In simpler terms, cotangent is not defined at the points \( x = n\pi \), where \( n \) is an integer.

Tangent

The tangent function, abbreviated as "tan," is another core trigonometric function. It relates the angle in a right triangle to the ratio of the length of the opposite side over the length of the adjacent side. In mathematical terms:

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

Understanding tangent through the unit circle involves seeing it as the slope of the line that runs from the origin through a given point on the circle.This interpretation helps comprehend its periodic nature and how it extends beyond the limits of a triangle.Tangent is undefined when the cosine of the angle is zero, as division by zero is not possible. These are the points at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Sine and Cosine Identities

Sine and cosine are the foundational building blocks of trigonometry. Many identities revolve around these two functions due to their integral role in defining other trigonometric functions. Key identities to know include:

• Using the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \)
• Double angle identities: \( \cos(2x) = \cos^2 x - \sin^2 x \) or \( \sin(2x) = 2 \sin x \cos x \)

Sine and cosine function as projections on the y-axis and x-axis, respectively, within the unit circle.They help define the position and arc of angles moving around the circle. This cyclic and symmetrical nature of sine and cosine leads to proofs and simplifications, like in our original exercise, using these identities.They simplify complex trigonometric expressions by transforming them into easier-to-handle sin and cos terms.

Common Denominator in Fractions

A common denominator is essential when adding or subtracting fractions. It is the least common multiple of the denominators of the fractions involved.For trigonometric identities involving fractions, such as our exercise, finding a common denominator allows the fractions to be combined into a single one.When dealing with symbolic fractions like those in trigonometry:

• For \( \frac{\cos x}{\sin x} \) and \( \frac{\sin x}{\cos x} \), the common denominator is \( \sin x \cos x \).
• Transform each term: \( \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x \cos x} \) and \( \frac{\sin x}{\cos x} = \frac{\sin^2 x}{\sin x \cos x} \).

This concept ensures that you can perform arithmetic operations efficiently and lead to simpler expressions.In the context of the exercise, it paved the way to apply identities and explore deeper relationships between the functions for proving the identity given.