Problem 79 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is proven: \( \frac{2(\tan x - \cot x)}{\tan^2 x - \cot^2 x} = \sin 2x \).

Step 1: Express in terms of Sine and Cosine

First, express the tangent and cotangent functions in terms of sine and cosine: \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). This will help simplify the identity.

Step 2: Simplify the Numerator

The numerator \( 2(\tan x - \cot x) \) becomes \[ 2 \left( \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} \right) = 2 \left( \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} \right) \] by getting a common denominator.

Step 3: Simplify the Denominator

The denominator \( \tan^2 x - \cot^2 x \) simplifies to \[ \left( \frac{\sin^2 x}{\cos^2 x} - \frac{\cos^2 x}{\sin^2 x} \right) = \frac{(\sin^4 x - \cos^4 x)}{\sin^2 x \cos^2 x} \] by expressing both terms with a common denominator.

Step 4: Use a Difference of Squares

The expression \( \sin^4 x - \cos^4 x \) can be written as \( (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \). Since \( \sin^2 x + \cos^2 x = 1 \), this simplifies to \( \sin^2 x - \cos^2 x \).

Step 5: Simplify the Entire Expression

Substitute the simplified numerator and denominator back into the original expression:\[ \frac{2(\sin^2 x - \cos^2 x)/(\sin x \cos x)}{(\sin^2 x - \cos^2 x)/(\sin^2 x \cos^2 x)} = \frac{2 \sin x \cos x} \] after canceling \( \sin^2 x - \cos^2 x \) in both numerator and denominator.

Step 6: Recognize the Double Angle Formula

The expression \( 2 \sin x \cos x \) is exactly \( \sin 2x \) due to the double angle identity \( \sin 2x = 2 \sin x \cos x \).

Key Concepts

Sine and Cosine

When exploring trigonometric identities, two fundamental functions that frequently appear are the sine (\( \sin x \)) and cosine (\( \cos x \)) functions. Understanding how to transition between various trigonometric functions with these two is crucial.

• Both sine and cosine functions help express other trigonometric functions. For instance, tangent (\( \tan x \)) is equal to \( \frac{\sin x}{\cos x} \), while cotangent (\( \cot x \)) is its reciprocal, given by \( \frac{\cos x}{\sin x} \).
• These expressions turn the more complex trigonometric expressions into simpler, more familiar forms that are often easier to manipulate, particularly when proofs or simplifications are required.
• A fundamental identity with these functions is \( \sin^2 x + \cos^2 x = 1 \), which acts as a foundational tool for many other trigonometric manipulations.

Recognizing these transformations and identities helps untangle complex expressions, paving the way for easier proofs and solutions.

Double Angle Formula

The Double Angle Formula is a powerful tool in trigonometry that simplifies expressions involving the angles of sine and cosine. Specifically, it helps relate an angle to its double, which can be invaluable in solving problems or proving identities.

• The formula for sine is: \( \sin 2x = 2 \sin x \cos x \).
• This formula comes into play when we have solutions like \( 2 \sin x \cos x \), which directly simplifies to \( \sin 2x \), reducing the complexity of an expression.
• This formula can greatly simplify calculations when manipulating equations by pulling out recognizable patterns and reducing the steps needed in proofs.
• Understanding and recognizing the Double Angle formula isn't just about brute memorization; it involves seeing past the initial complexity of an equation and knowing that such formulas exist to bridge gaps in trigonometric transformations.

In our solution, recognizing \( 2 \sin x \cos x \) as \( \sin 2x \) allows us to complete the proof with ease, reflecting the formula's essential role.

Difference of Squares

The Difference of Squares is an algebraic technique that simplifies products of expressions that are structured a certain way. This technique is especially helpful when dealing with trigonometric identities like our problem.

• The basic form of Difference of Squares is \( a^2 - b^2 = (a-b)(a+b) \), which can simplify otherwise challenging expressions.
• In trigonometry, it is particularly useful when the expressions inside resemble squares, like \( \sin^4 x - \cos^4 x \).
• This expression can be rewritten as \( (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \), where \( \sin^2 x + \cos^2 x = 1 \) simplifies it further to \( \sin^2 x - \cos^2 x \).
• This helps ease simplification by reducing the expression to a single term that we can further manipulate, as seen in many steps in solving the original exercise.

The Difference of Squares is a prime example of how algebraic identities can assist in trigonometric problem-solving, turning seemingly complex operations into manageable steps.