Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Identity verified; both sides are equal.

Step 1: Understand the Identity

We need to verify that the expression on the left-hand side (LHS), \(\frac{1-\sin x}{1+\sin x}\), is equal to the expression on the right-hand side (RHS), \((\sec x-\tan x)^{2}\).

Step 2: Simplify the RHS

Let's simplify the RHS. \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). Therefore, \(\sec x - \tan x = \frac{1}{\cos x} - \frac{\sin x}{\cos x} = \frac{1 - \sin x}{\cos x}\). Now, square this expression to get:\[(\sec x - \tan x)^2 = \left(\frac{1 - \sin x}{\cos x}\right)^2 = \frac{(1 - \sin x)^2}{\cos^2 x}\].

Step 3: Simplify and Compare Both Sides

Now, we simplify the LHS:\( \frac{1 - \sin x}{1 + \sin x} \cdot \frac{1 - \sin x}{1 - \sin x} = \frac{1 - 2\sin x + \sin^2 x}{\cos^2 x}\)This is equal to:\[\frac{(1 - \sin x)^2}{\cos^2 x}\].The simplified form of both equations is identical, confirming the identity.

Key Concepts

Secant Function

The secant function is a fundamental component of trigonometry that you encounter when dealing with trigonometric identities. Mathematically, it is denoted as \( \sec x \) and is defined as the reciprocal of the cosine function:

• \( \sec x = \frac{1}{\cos x} \)

This means that the secant function takes the cosine of an angle and flips it. The secant function is particularly useful for simplifying expressions where division by cosine occurs. By introducing \( \sec x \), equations can often be rewritten in a form that is easier to manipulate or solve.
For instance, in the given identity exercise, expressing \( \sec x \) as \( \frac{1}{\cos x} \) helped us simplify the right-hand side of the equation. Breaking down expressions into functions like secant is a common step in verifying and proving trigonometric identities.

Tangent Function

The tangent function is another key trigonometric identity often found in math problems. It is represented as \( \tan x \) and relates to the sine and cosine functions:

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

Understanding \( \tan x \) is important because it often appears in various trigonometric identities, including simplification tasks. The tangent function is helpful in expressions and equations where both sine and cosine terms are present. It allows their relationship to be clearly understood and simplified.
In our original exercise, converting \( \tan x \) to its derived form played a crucial role in simplifying the identity. Bringing together secant and tangent functions into one framework enables us to show equivalency in many trigonometric identities.

Trigonometric Simplification

Trigonometric simplification involves converting complex trigonometric expressions into simpler or more manageable forms. This process often uses fundamental identities, such as:

• Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal identities: \( \sec x = \frac{1}{\cos x} \), \( \tan x = \frac{\sin x}{\cos x} \)

In trigonometric simplification, you're typically trying to make the expression simpler to read or verify an identity like the one presented in our exercise.
To simplify the given identity, we rephrased both sides using trigonometric identities like secant and tangent. We also used basic algebraic techniques such as squaring and factoring. By comparing both sides after simplification, we confirmed their equivalency.
Trigonometric simplification is a powerful tool to make seemingly complex equations easier to understand and solve, which is why it's essential in verifying trigonometric identities.