Problem 74 - Algebra and Trigonometry

Question

Rewrite each expression in terms of the given function or functions. \(\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x} ; \sec x\) and \(\tan x\)

Step-by-Step Solution

Answer

The expression rewritten in terms of sec(x) and tan(x) is \(-4 \cdot \sec^2 x \cdot \tan x\)

Step 1: Start with the given expression

We are starting with \(\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x}\)

Step 2: Combine the fractions

To combine these fractions, we should find common denominators. For these fractions, the common denominator will be \((1-\sin x)(1+\sin x)\) which is \((1-\sin^2 x)\) because of the identity \(a^2-b^2 = (a-b)(a+b)\). The resulting expression will be \(\frac{(1-\sin x)^2 - (1+\sin x)^2}{1-\sin^2 x}\)

Step 3: Recognize the identity

From the denominator of the resulting fraction, knowing that \(1-\sin^2 x = cos^2 x\) is crucial here. So, we substitute \(cos^2 x\) instead of \(1-\sin^2 x\) in the denominator. Now we have \(\frac{(1-\sin x)^2 - (1+\sin x)^2}{cos^2 x}\)

Step 4: Simplify the numerator

After simplifying, we get \(\frac{-4 sin x}{cos^2 x}\)

Step 5: Convert to the desired functions

Finally we can convert this to the desired functions \(\sec x\) and \(\tan x\). We know that \(\sec x = \frac{1}{cos x}\) and \(\tan x = \frac{sin x}{cos x}\). Thus our expression becomes \(-4 \cdot \sec^2 x \cdot \tan x\)

Key Concepts

Secant Function

The secant function is a fundamental trigonometric ratio, often denoted as \( \sec x \). It is defined in terms of the cosine function, specifically as the reciprocal of cosine. In mathematical terms, this is expressed as:

• \( \sec x = \frac{1}{\cos x} \)

The secant function is essential for understanding various trigonometric relationships and identities. It helps to convert expressions involving reciprocal trigonometric functions into more manageable forms.
In the given problem, the expression \(-4 \cdot \sec^2 x \cdot \tan x\) involves the secant function squared, \( \sec^2 x \), which is simply \( \left(\frac{1}{\cos x}\right)^2 \).
This shows how secant transforms the original trigonometric expressions into others using its reciprocal property.

Tangent Function

The tangent function, denoted as \( \tan x \), is another basic trigonometric function. It is defined as the ratio of the sine and cosine functions:

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

The tangent function is widely used to express trigonometric relationships and solve various kinds of problems.
For the expression \(-4 \cdot \sec^2 x \cdot \tan x\) in the exercise, \( \tan x \) plays a critical role since it transforms the original subtraction of fractions involving sines and cosines into a single form using tangent and secant. Chaining functions this way simplifies complex trigonometric expressions.

Trigonometric Expressions

Trigonometric expressions involve the basic trigonometric functions like sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent). They underpin various types of mathematical problems, especially those dealing with angles and triangles.
Combining or manipulating these expressions often requires employing identities, such as:

• Pythagorean Identities: \( \sin^2 x + \cos^2 x = 1 \)
• Tangent and Secant Identities

Because trigonometric expressions can be lengthy or complex, rewriting them using different functions simplifies calculations.
In our problem, rewriting \( \frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x} \) into a form that involves \( \sec x \) and \( \tan x \) simplifies addressing the identity \( \tan^2 x + 1 = \sec^2 x \), which relates both functions. This transformation not only simplifies the expression but also makes it easier to understand the underlying relationships between the trigonometric functions.