Problem 66 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\cot 2 x=\frac{1-\tan ^{2} x}{2 \tan x}$$

Step-by-Step Solution

Answer

Identity is verified as correct.

Step 1: Recall Trigonometric Identity

We start by recalling the identity for \( \cot 2x \), which is the cotangent of a double angle. The identity is given by:\[ \cot 2x = \frac{1 - \tan^2 x}{2 \tan x} \]This is what we need to prove is equal to \( \frac{1 - \tan^2 x}{2 \tan x} \). We will verify this equation step by step.

Step 2: Use Double Angle Identity for Tangent

The double angle identity for tangent is:\[ \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x} \]Knowing that \( \cot \theta = \frac{1}{\tan \theta} \), the cotangent double angle identity is:\[ \cot(2x) = \frac{1}{\tan(2x)} = \frac{1 - \tan^2 x}{2 \tan x} \]

Step 3: Verify and Simplify

To verify, find the expression \( \cot(2x) \) using the identity derived.Since \( \cot(2x) = \frac{1}{\tan(2x)} \), substituting the expression for \( \tan(2x) \), we have:\[ \cot(2x) = \frac{1}{\frac{2 \tan x}{1 - \tan^2 x}} = \frac{1 - \tan^2 x}{2 \tan x} \]This confirms the given identity.

Key Concepts

Understanding Cotangent

Cotangent, often written as \( \cot \), is one of the six fundamental trigonometric functions. It is closely related to the tangent function. In terms of a right triangle, cotangent for an angle \( \theta \) is defined as the ratio of the adjacent side to the opposite side. Mathematically, this can be represented as:

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

Understanding \( \cot \) in this way can help to visualize its role in trigonometry, especially in relation to tangent. Cotangent is helpful when considering angles and their complements, as well as in complex calculations like double angles.
In the example problem, \( \cot(2x) \) was derived to establish a specific identity. Using known identities and step-by-step verification helps to fully grasp the concept of cotangent and how it interlinks with other trigonometric functions.

Exploring Double Angle Formulas

Double angle formulas are essential tools in trigonometry, used to simplify expressions involving angles that are doubled. For sine, cosine, and tangent, these formulas help reveal relationships within trigonometric functions when the angle is multiplied by two.
Here are the common double angle formulas:

• \( \sin(2x) = 2 \sin x \cos x \)
• \( \cos(2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x \)
• \( \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}\)

For the exercise at hand, knowing \( \tan(2x) \) is crucial in proving the cotangent identity. By substituting the double angle formula for tangent into the expression for cotangent, we simplify and confirm the identity. It’s this manipulation of known formulas that highlights the power of double angle formulas in trigonometry. Understanding these can greatly increase your efficiency in solving complex problems.

The Role of Tangent in Trigonometry

Tangent, like sine and cosine, is a fundamental trigonometric function. It’s usually defined as the ratio of the sine to the cosine of an angle, \( \theta \). In the context of a right triangle, \( \tan \theta \) can be described as the ratio of the opposite side to the adjacent side:

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

This function is vital in both understanding angular relationships and in manipulating equations through identities. In our identity problem, \( \tan x \) helps frame the double angle identity and its inverse with cotangent.
Proper utilization of tangent is necessary for the transition from the expression of one trigonometric function to another. This manipulation allows for the proof of identities, such as the one provided in the exercise. Recognizing the interplay between tangent and cotangent, especially when working with double angles, is key to mastering these concepts.