Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \sec x(\sec x+\tan x) \).

Step 1: Identify the Identity to be Verified

We need to verify the identity \( \frac{\sec x}{\sec x-\tan x}=\sec x(\sec x+\tan x) \). This means we have to prove that the left-hand side (LHS) equals the right-hand side (RHS).

Step 2: Simplify the LHS

Start with the left-hand side: \( \frac{\sec x}{\sec x-\tan x} \). Multiply numerator and denominator by the conjugate of the denominator \( \sec x+\tan x \):\[\frac{\sec x}{\sec x-\tan x} \cdot \frac{\sec x+\tan x}{\sec x+\tan x} = \frac{\sec x (\sec x + \tan x)}{(\sec x)^2 - (\tan x)^2}\]

Step 3: Use Pythagorean Identity

In the previous step's denominator, use the Pythagorean identity \( \sec^2 x - \tan^2 x = 1 \):\[\frac{\sec x (\sec x + \tan x)}{1} = \sec x (\sec x + \tan x)\]

Step 4: Compare with RHS

The expression \( \sec x (\sec x + \tan x) \) matches exactly with the right-hand side. Therefore, the identity is verified.

Key Concepts

Right-Hand Side (RHS)

In mathematics, the right-hand side (RHS) of an equation refers to the part of the equation that appears to the right of the equal sign. For any equation like our trigonometric identity, the RHS needs to be shown equivalent to the left-hand side (LHS) for the identity to be verified.
Let's consider our exercise: After transforming and simplifying the LHS, you compare it to the RHS:

• The RHS in this context is \( \sec x(\sec x+\tan x) \).
• Once we've applied algebraic manipulations to the LHS, we want our results to look exactly like the RHS.

Verification in such problems is the core step, as it requires not only manipulation skills but also a deep understanding of the identities involved. By comparing the manipulated LHS to the RHS, and confirming they are indeed equal, the identity is verified.

Left-Hand Side (LHS)

The left-hand side (LHS) of an equation represents the expression or terms on the left side of the equal sign. In verifying trigonometric identities, we often manipulate the LHS to simplify it and make it look like the RHS.
In our given identity verification, we start with the LHS:

• \( \frac{\sec x}{\sec x-\tan x} \)
• The goal is to use mathematical operations, potentially including factoring, adding, subtracting, or multiplying, to transform this side until it matches the RHS.
• Here, by multiplying the numerator and denominator by the conjugate \( \sec x+\tan x \), and simplifying using identities, we successfully transform the LHS.

This methodical approach is crucial for verifying identities, ensuring that both sides are equal without altering the equation's intrinsic properties.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry derived from the Pythagorean theorem. One of the primary forms used in trigonometric transformations and simplifications is:

• \( \sec^2 x - \tan^2 x = 1 \)

In our problem-solving context:

• This identity is applied to the denominator after multiplying with the conjugate, simplifying \( (\sec x)^2 - (\tan x)^2 \) to 1.
• Recognizing and using this identity help transform complex trigonometric expressions into simpler forms.
• It plays a key role not just in this exercise, but in many others involving trigonometric identities.

Understanding these identities allows you to tackle complex trigonometric functions with ease, ensuring accurate simplifications and transformations.