Problem 65 - Algebra and Trigonometry

Question

59–76 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 as \( \sin 2x = \frac{2(\tan x - \cot x)}{\tan^2 x - \cot^2 x} \).

Step 1: Recall Trigonometric Identities

To begin, recall the definitions of tangent and cotangent: \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Additionally, remember the double angle identity for sine: \( \sin(2x) = 2 \sin x \cos x \).

Step 2: Rewrite the Numerator

Rewrite the numerator \( 2(\tan x - \cot x) \) using the definitions of \( \tan x \) and \( \cot x \): \[2\left( \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} \right) = 2 \left( \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} \right)\].

Step 3: Rewrite the Denominator

Rewrite the denominator \( \tan^2 x - \cot^2 x \) as follows: \[\left( \frac{\sin^2 x}{\cos^2 x} - \frac{\cos^2 x}{\sin^2 x} \right) = \frac{\sin^4 x - \cos^4 x}{\sin^2 x \cos^2 x}\].

Step 4: Simplify the Expression

Substitute the expressions for the numerator and denominator into the left-hand side of the identity: \[ \frac{2 \left( \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} \right)}{\frac{\sin^4 x - \cos^4 x}{\sin^2 x \cos^2 x}} \].This simplifies to: \[ \frac{2(\sin^2 x - \cos^2 x)}{\sin^4 x - \cos^4 x} \times \frac{\sin^2 x \cos^2 x}{\sin x \cos x} \].

Step 5: Simplify Further Using a Trigonometric Identity

Notice \( \sin^4 x - \cos^4 x \) can be factored using the difference of squares: \[ (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)\].Since \( \sin^2 x + \cos^2 x = 1 \), it simplifies to \( \sin^2 x - \cos^2 x \).

Step 6: Cancel Terms and Conclude

Now, cancel \( (\sin^2 x - \cos^2 x) \) from both the numerator and denominator:\[ \frac{2}{\sin x \cos x} \]. This yields \( \frac{2}{\sin x \cos x} = 2 \sin x \cos x \), which is precisely \( \sin(2x) \). Thus, the identity \( \frac{2(\tan x - \cot x)}{\tan^2 x - \cot^2 x} = \sin 2x \) is verified.

Key Concepts

Tangent and Its Properties

Tangent, often abbreviated as \( \tan \), is a fundamental trigonometric function that expresses the ratio of the side opposite an angle to the side adjacent to that angle in a right triangle. The formula for tangent using sine and cosine is: \[ \tan x = \frac{\sin x}{\cos x} \] The function is periodic, with a period of \( \pi \), and is undefined for angles where \( \cos x = 0 \) (which happen at odd multiples of \( \frac{\pi}{2} \)).

• Periodic Behavior: Tangent repeats its values every \( \pi \).
• Asymptotes: Occur where cosine is zero, resulting in vertical asymptotes on its graph.
• Range: Tangent values can extend from negative infinity to positive infinity.

Understanding these properties is crucial when solving trigonometric identities involving tangent, as seen in problems like this one.

Cotangent and Its Relationship to Tangent

Cotangent, symbolized as \( \cot \), is another prominent trigonometric function. It's essentially the reciprocal of the tangent function, defined by the ratio of the adjacent side to the opposite side of a right triangle: \[ \cot x = \frac{\cos x}{\sin x} \] Just like tangent, cotangent is periodic and also has undefined points, specifically where \( \sin x = 0 \) (any multiple of \( \pi \)).

• Period: The cotangent function repeats every \( \pi \), similar to tangent.
• Undefined Points: Occur where \( \sin x = 0 \). These correspond to the zeros of the sine function.
• Range: The values range from negative to positive infinity.

In our exercise, understanding cotangent's properties allows us to rewrite and manipulate expressions, such as turning differences into simpler fractions for simplification.

Understanding the Double Angle Identity for Sine

The double angle identity is a valuable tool in trigonometry that expresses trigonometric functions of two angles. For sine, the double angle identity is: \[ \sin(2x) = 2 \sin x \cos x \] This formula is particularly useful when simplifying trigonometric expressions or when solving equations involving double angles.

• Utility: It is used to reflect the value of sine for double the value of an angle \( x \).
• Simplification: Helps convert complex expressions that involve angles like \( 2x \) into products of basic sine and cosine functions.
• Verification: As shown in the exercise, verifying identities such as \( \frac{2(\tan x - \cot x)}{\tan^2 x - \cot^2 x} = \sin 2x \) involves recognizing and utilizing this identity at crucial steps.

Through the exercise, the double angle identity helps verify the trigonometric identity by simplifying expressions to match common forms like \( 2 \sin x \cos x \), confirming the final simplified result.