Problem 44 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1-\sin x}{1+\sin x}=(\sec x-\tan x)^{2}$$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Simplify the Right Side

First, we'll simplify the right side of the equation, \((\sec x - \tan x)^{2}\).Recall that \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). Substituting these into the expression, we have:\((\frac{1}{\cos x} - \frac{\sin x}{\cos x})^{2} = \left(\frac{1 - \sin x}{\cos x}\right)^{2}.\)Expanding this, we get:\(\frac{(1 - \sin x)^{2}}{\cos^{2} x}\).The expression simplifies to:\(\frac{1 - 2\sin x + \sin^{2} x}{\cos^{2} x}.\)

Step 2: Use Pythagorean Identity

Now, apply the Pythagorean identity \(1 - \sin^{2} x = \cos^{2} x\) to simplify the numerator.Rewriting the numerator:\(1 - 2\sin x + \sin^{2} x = (1 - \sin^{2} x) - 2\sin x\). Since \(1 - \sin^{2} x\) is equal to \(\cos^{2} x\), substitute it back:\(\cos^{2} x - 2\sin x\).This gives us \(\frac{\cos^{2} x - 2\sin x}{\cos^{2} x}.\)

Step 3: Simplify to Match the Left Side

Now, let's simplify the right side further: \[\frac{\cos^{2} x - 2\sin x}{\cos^{2} x} = \frac{1 - \sin x}{1 + \sin x}\].You know that:\(\frac{1 - \sin x}{\cos^{2} x}\) can be rewritten by multiplying the numerator and the denominator by its conjugate: \(\left(\frac{1 - \sin x}{1 + \sin x}\right)\left(\frac{1 + \sin x}{1 + \sin x}\right)\).Thus, the expression becomes:\(\frac{1 - \sin^2 x}{(1 + \sin x)^{2}}\).Since \(1 - \sin^2 x = \cos^2 x\), we get:\(\frac{\cos^2 x}{(1 + \sin x)^{2}} = \frac{1 - \sin x}{1 + \sin x}\).Now both sides are equal, verifying the identity.

Key Concepts

Secant Function

The secant function, often abbreviated as \( \sec x \), is closely related to the cosine function. In fact, the secant function is the reciprocal of the cosine, which means:

• \( \sec x = \frac{1}{\cos x} \)
This relationship is an integral part of trigonometry and helps in solving trigonometric identities. Understanding secant aids in simplifying expressions where cosine appears in the denominator. For example, in the exercise, transforming the secant term into its equivalent expression using cosine allows us to work with algebraic manipulations more efficiently. This conversion is often the first step when verifying trigonometric identities to find common ground with other trigonometric functions.

Tangent Function

The tangent function, denoted as \( \tan x \), is another trigonometric function that frequently comes up in identities and equations. It's defined as the ratio of the sine and cosine functions:

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

The tangent function provides a way to relate the sine and cosine, especially when you're working with right triangles in the context of the unit circle. In our exercise, expressing the tangent function in terms of sine and cosine makes it easier to manipulate equations. By substituting \( \tan x \) with its definition in the initial expression, we can align it with other parts of the problem, thereby simplifying the process of verifying trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. One form of this identity is:

• \( 1 - \sin^2 x = \cos^2 x \)

This identity states that the sum of the square of sine and cosine of any angle is always one. For verifying identities, this relationship is extremely useful for transforming parts of an expression. It allows us to replace \( 1 - \sin^2 x \) with \( \cos^2 x \) or vice versa. Applying this identity in the step-by-step solution simplifies the numerical part of the numerator, allowing the expression to be rewritten in a more straightforward form. Therefore, this identity is crucial for breaking down complex trigonometric expressions into simpler components that can be more easily managed and solved.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make problems easier to solve. This skill is vital not only in algebra but also in calculus and trigonometry. During the solution of our exercise, several algebraic techniques were utilized such as:

• Factoring expressions
• Distributing terms
• Multiplying by conjugates

These techniques help in changing an expression into a more workable form. For instance, in the exercise, multiplying by the conjugate helps in alleviating complex fractions allowing them to become more pliable. These manipulations often lead to a situation where matching both sides of an identity becomes apparent and straightforward. Mastery of algebraic manipulation thus serves as an invaluable tool when dealing with trigonometric identities and is a crucial part of simplifying and verifying expressions.