Problem 80 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \cot 2 x=\frac{1-\tan ^{2} x}{2 \tan x} $$

Step-by-Step Solution

Answer

The identity is proved using double angle formulas and trigonometric identities.

Step 1: Understanding the Identity

We need to prove that \(\cot(2x) = \frac{1 - \tan^2 x}{2 \tan x}\). Recall that \(\cot(2x) = \frac{\cos(2x)}{\sin(2x)}\). Therefore, we need to express both \(\cos(2x)\) and \(\sin(2x)\) in terms of \(\tan(x)\).

Step 2: Express \(\cos(2x)\) in Terms of \(\tan x\)

Using the double angle formula for cosine, \(\cos(2x) = 1 - 2\sin^2 x\). Also, since \(\tan x = \frac{\sin x}{\cos x}\), we can express \(\sin^2 x = \frac{\tan^2 x}{1 + \tan^2 x}\) leading to \(\cos(2x) = 1 - \frac{2\tan^2 x}{1+\tan^2 x}\).

Step 3: Express \(\sin(2x)\) in Terms of \(\tan x\)

Using the double angle formula for sine, \(\sin(2x) = 2\sin x \cos x\). Since \(\tan x = \frac{\sin x}{\cos x}\), we can rewrite this as \(\sin(2x) = \frac{2\tan x}{1 + \tan^2 x}\).

Step 4: Combine to Find \(\cot(2x)\)

Now substitute \(\sin(2x)\) and \(\cos(2x)\) in \(\cot(2x) = \frac{\cos(2x)}{\sin(2x)}\):\[\cot(2x) = \frac{1 - \frac{2\tan^2 x}{1+\tan^2 x}}{\frac{2\tan x}{1+\tan^2 x}}\].

Step 5: Simplify the Expression

Simplify the fraction above:\[\cot(2x) = \frac{(1 + \tan^2 x) - 2\tan^2 x}{2\tan x}\] \[= \frac{1 - \tan^2 x}{2\tan x}\], which is the identity we needed to prove.

Key Concepts

Cotangent

The cotangent function, often abbreviated as \( \cot \), is one of the fundamental trigonometric functions closely related to the tangent function. Understanding cotangent begins with its definition: it is the reciprocal of the tangent function, specifically, \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \). This means it represents the ratio of the adjacent side to the opposite side in a right triangle when considering an angle \( x \).

Key properties of cotangent to remember include:

• Cotangent is undefined where \( \sin x = 0 \), just like tangent is undefined where \( \cos x = 0 \).
• It shares periodicity with tangent, repeating every \( \pi \) radians.
• The cotangent function is useful in simplifying and solving various trigonometric identities and equations, especially when dealing with angles multiplied or divided by a factor.

Understanding these properties is crucial when working with equations that require manipulation using reciprocal and inverse relationships in trigonometry.

Double Angle Formulas

Double angle formulas are key identities in trigonometry that allow us to express trigonometric functions of double angles \( 2x \) in terms of functions of the angle \( x \). These expressions are particularly useful when proving identities and solving trigonometric equations.

The most common double angle formulas include:

• \( \cos(2x) = \cos^2 x - \sin^2 x \)
• Or alternatively, \( \cos(2x) = 1 - 2\sin^2 x \) and \( \cos(2x) = 2\cos^2 x - 1 \)
• \( \sin(2x) = 2\sin x \cos x \)

When dealing with tangent, substitute these forms using the identity \( \tan x = \frac{\sin x}{\cos x} \) to simplify expressions.

These formulas help bridge single and double angles, facilitating conversions that lead to simplifying complex trigonometric expressions. In problems, they are often used to express angles as a sum or difference, making it easier to factor and solve trigonometric equations.

Tangent

The tangent function, expressed as \( \tan x \), is another vital component of trigonometry, representing the ratio between the opposite and adjacent sides in a right triangle relative to an angle \( x \). The tangent function is key in converting and manipulating trigonometric identities because of its reciprocal nature and close relation with sine and cosine.

Key aspects of tangent to keep in mind include:

• \( \tan x = \frac{\sin x}{\cos x} \), showcasing its dependence on sine and cosine.
• Tangent has vertical asymptotes where cosine equals zero, which occur at angles of \( \frac{\pi}{2} + n\pi \).
• The tangent function is periodic with a period of \( \pi \) radians, meaning it repeats its values every \( \pi \) intervals.

It's crucial in problems that require expressions in terms of simple identities or combined with cotangent and other reciprocal ratio functions.

By understanding the tangent function's properties, students gain tools for simplifying, rearranging, and ultimately solving complex trigonometric identities and equations. This function is often pivotal in problems dealing with tangent and cotangent relationships, such as the double angle identity shown in this exercise.