Problem 80 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x$$

Step-by-Step Solution

Answer

The identity is verified: both sides equal \(4 \tan x \sec x\).

Step 1: Establish a Common Denominator

The given expression is \( \frac{1+\sin x}{1-\sin x} - \frac{1-\sin x}{1+\sin x} \). To simplify this, we first establish a common denominator, which is \((1-\sin x)(1+\sin x)\). This is equivalent to \(1-\sin^2 x\), which simplifies to \(\cos^2 x\).

Step 2: Expand the Expression

Rewrite each term with the common denominator: \(\frac{(1+\sin x)^2 - (1-\sin x)^2}{\cos^2 x}\). Use the identity \((a^2 - b^2) = (a-b)(a+b)\) to simplify. Here, after expansion: \[((1 + \sin x) - (1 - \sin x))((1 + \sin x) + (1 - \sin x)) = (2\sin x)\cdot2 = 4\sin x\].

Step 3: Simplify to a Single Trigonometric Function

Now the expression simplifies to \(\frac{4\sin x}{\cos^2 x}\). This can be rewritten as \(4 \cdot \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = 4 \tan x \sec x\).

Step 4: Validate the Identity

Both sides of the equation \(\frac{1+\sin x}{1-\sin x} - \frac{1-\sin x}{1+\sin x} = 4 \tan x \sec x \) are equal, showing the identity holds true.

Key Concepts

Common Denominator

When working with fractions in algebra or trigonometry, finding a common denominator is crucial to simplify the expression. In this exercise, we deal with the fractions \( \frac{1 + \sin x}{1 - \sin x} \) and \( \frac{1 - \sin x}{1 + \sin x} \). To subtract these fractions easily, both need to have the same denominator.

The common denominator for these fractions is found by multiplying the individual denominators together:

• \((1 - \sin x)(1 + \sin x)\)

This is recognized as a difference of squares formula: \(1 - \sin^2 x\). This formula is part of the Pythagorean identities, and it simplifies to \(\cos^2 x\) because \(\cos^2 x + \sin^2 x = 1\).

By converting both fractions to have \(\cos^2 x\) as a denominator, it becomes easier to combine and simplify the expression. Remembering that achieving a common denominator is the key step to transforming complex fractional expressions into manageable ones can be very helpful.

Trigonometric Simplification

Trigonometric simplification is the process of using identities and basic algebraic techniques to condense trigonometric expressions. After establishing a common denominator of \( \cos^2 x \), we rewrite the given expression:

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

This uses the difference of squares formula \((a^2 - b^2) = (a-b)(a+b)\). Here:

• The expression becomes \((2\sin x)(2) = 4\sin x\)

The simplified expression is \(\frac{4 \sin x}{\cos^2 x}\).

To further simplify, recognize that \(\frac{\sin x}{\cos^2 x}\) can be rewritten using other trigonometric identities. We know:

• \(\frac{\sin x}{\cos x} = \tan x\)
• \(\frac{1}{\cos x} = \sec x\)

By applying these identities, the expression \(4 \cdot \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}\) simplifies directly to \(4 \tan x \sec x\). Simplification helps in recognizing broader trigonometric relationships.

Difference of Squares

The difference of squares is a formula used in algebra to simplify expressions, and it appears frequently in trigonometry as well. The formula \((a^2 - b^2) = (a-b)(a+b)\) works by breaking down expressions into simpler multiplication terms.
In this exercise, the expression \((1 + \sin x)^2 - (1 - \sin x)^2\) uses this principle. It transforms into:

• \((1 + \sin x) - (1 - \sin x) = 2\sin x\)
• \((1 + \sin x) + (1 - \sin x) = 2\)

Now, multiply these results: \((2\sin x) \cdot 2 = 4\sin x\).

This effectively reduces a complex expression into something more manageable, showing the importance and power of recognizing potential identities when simplifying trigonometric equations. The difference of squares helps in reducing expressions quickly and efficiently, which is a key skill in solving many types of mathematical problems.