Problem 57 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{2 \tan x}{1+\tan ^{2} x}=\sin 2 x $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{2 \tan x}{1 + \tan^2 x} = \sin 2x \).

Step 1: Start with the left side of the identity

We begin with the expression on the left side of the identity: \( \frac{2 \tan x}{1 + \tan^2 x} \). Our goal is to manipulate this expression to show that it is equivalent to \( \sin 2x \).

Step 2: Use the double angle identity for sine

Recall the double angle identity for sine, which is \( \sin 2x = 2 \sin x \cos x \). We will aim to express \( \frac{2 \tan x}{1 + \tan^2 x} \) in this form.

Step 3: Recall identities and express in terms of sine and cosine

We know that \( \tan x = \frac{\sin x}{\cos x} \). Substitute this into the expression: \[ \frac{2 \left( \frac{\sin x}{\cos x} \right)}{1 + \left( \frac{\sin x}{\cos x} \right)^2} = \frac{2 \left( \frac{\sin x}{\cos x} \right)}{1 + \frac{\sin^2 x}{\cos^2 x}} \]

Step 4: Simplify the denominator

Simplify the denominator: \[ 1 + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \]Using the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), the denominator becomes \( \frac{1}{\cos^2 x} \).

Step 5: Simplify the entire expression

Substitute back into the expression: \[ \frac{2 \left( \frac{\sin x}{\cos x} \right)}{\frac{1}{\cos^2 x}} = 2 \sin x \cos x \] This shows that \( \frac{2 \tan x}{1 + \tan^2 x} = \sin 2x \).

Step 6: Conclude the proof

We have shown that both sides of the given identity are equal. Therefore, the identity \( \frac{2 \tan x}{1 + \tan^2 x} = \sin 2x \) is verified.

Key Concepts

Double Angle Identity

The double angle identity is a powerful tool in trigonometry. It's particularly useful when simplifying expressions and solving equations. The double angle identity for sine states:

• \( \sin 2x = 2 \sin x \cos x \)

This identity helps us express a trigonometric function of a double angle in terms of sine and cosine. By doing this, we can often transform complex expressions into simpler forms.
In the given problem, we started with the expression \( \frac{2 \tan x}{1 + \tan^2 x} \). To prove this equals \( \sin 2x \), we leveraged the double angle identity. Through substitution and algebraic manipulation, we simplified the expression to match \( 2 \sin x \cos x \), thus verifying the identity.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the trigonometric functions sine and cosine. It is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is a cornerstone for transforming and simplifying trigonometric expressions. It comes in handy especially when dealing with squared terms.
In our verification process, this identity allowed us to simplify the denominator \( 1 + \frac{\sin^2 x}{\cos^2 x} \) in the fraction \( \frac{2 \left( \frac{\sin x}{\cos x} \right)}{1 + \frac{\sin^2 x}{\cos^2 x}} \). By recognizing \( \cos^2 x + \sin^2 x = 1 \), we transformed the denominator to \( \frac{1}{\cos^2 x} \). This was a crucial step towards simplifying the expression and ultimately matching it with \( \sin 2x \).

Trigonometric Functions

Trigonometric functions are essential to describing the relationships in a right-angled triangle. The basic functions include sine, cosine, and tangent. Here are key relationships:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

In the provided exercise, understanding the tangent function as \( \tan x = \frac{\sin x}{\cos x} \) was vital. This allowed us to convert the tangent in the original equation into terms of sine and cosine. Converting functions into sine and cosine terms is often an effective strategy when dealing with identities and when our target expression is in terms of these basic trigonometric functions.
Recognizing these functions and identities not only aids in solving equations but also provides a deeper understanding of the nature of trigonometric relationships.