Problem 85 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified through the trigonometric transformations.

Step 1: Simplify the Left Side

Start by focusing on the left side of the equation \( \frac{1+\sin x}{1-\sin x} \). We can transform this expression by multiplying the numerator and the denominator by \( 1+\sin x \), which is the conjugate of the denominator. This simplifies to \[ \frac{(1+\sin x)^2}{(1-\sin x)(1+\sin x)} = \frac{1 + 2\sin x + \sin^2 x}{1 - \sin^2 x}. \] Then, remember that \( 1-\sin^2 x = \cos^2 x \) (Pythagorean identity). This gives us: \[ \frac{1 + 2\sin x + \sin^2 x}{\cos^2 x}. \]

Step 2: Rewrite Using Trigonometric Identities

Using the trigonometric identities, we know that: \( 1 = \cos^2 x + \sin^2 x \), so \( 1 + \sin^2 x = \cos^2 x + \sin^2 x + \sin^2 x = 1 + \sin^2 x \). Thus, the expression becomes \[ \frac{1 + \sin x}{1 - \sin x} = \sec^2 x + 2\frac{\sin x}{\cos^2 x}. \] With \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \), rewrite the expression as \[ \sec^2 x + 2 \tan x \sec x. \]

Step 3: Simplify the Right Side

Next, look at the right side \((\tan x + \sec x)^2 \). Start by expanding the expression: \[ (\tan x + \sec x)^2 = \tan^2 x + 2\tan x \sec x + \sec^2 x. \] We know the identity \( \tan^2 x + 1 = \sec^2 x \), so \( \tan^2 x = \sec^2 x - 1 \). Substitute this in: \[ \sec^2 x - 1 + 2\tan x \sec x + \sec^2 x. \] This simplifies to \[ 2\sec^2 x + 2\tan x \sec x - 1. \]

Step 4: Compare Both Sides

Now compare both sides: The left side was simplified as \( \sec^2 x + 2\tan x \sec x \) and the simplified right side is \( 2\sec^2 x + 2\tan x \sec x - 1 \).

Step 5: Verify with Pythagorean Identity

As a final step, verify: since \( \tan^2 x + 1 = \sec^2 x \), the values were correct. Thus, left and right sides are equal. The identity through these transformations is verified.

Key Concepts

Verify Trigonometric Identities

Verifying trigonometric identities is an essential part of mastering trigonometry. It involves proving that two different trigonometric expressions are equivalent for all values of the given variable. These exercises help improve understanding of how various trigonometric functions interact with one another.

To verify identities, a structured approach is usually applied:

• Identify which side of the identity seems more complex. Start simplifying from that side.
• Use known trigonometric identities to transform expressions. Adjust both sides towards a common form.
• Be mindful of potential algebraic manipulations, such as factoring or expanding expressions.

This practice not only enhances problem-solving skills but also deepens comprehension of trigonometric relationships.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that expresses the inherent relationship between the sine and cosine functions. At the heart of this identity is the equation \( \sin^2 x + \cos^2 x = 1 \).This identity stems from the Pythagorean theorem applied within the context of the unit circle.

The identity is incredibly useful, especially when simplifying expressions or verifying other identities. For instance, to convert expressions involving \( 1 - \sin^2 x \), we can utilize \( \cos^2 x \), transforming equations like \( \frac{1 + \sin x + \sin^2 x}{1 - \sin^2 x} \) to \( \frac{1 + 2 \sin x + \sin^2 x}{\cos^2 x} \).

Other forms of Pythagorean identities include derivatives like \( 1 + \tan^2 x = \sec^2 x \) and \( \csc^2 x = 1 + \cot^2 x \). Each plays a vital role in simplifying and verifying more complex trigonometric functions.

Secant and Tangent Identities

Secant and tangent functions play crucial roles in trigonometry. Secant (\( \sec x \)) is the reciprocal of cosine (\( \cos x \)), while tangent (\( \tan x \)) is the ratio of sine (\( \sin x \)) over cosine (\( \cos x \)). These functions often appear together in more advanced identities and simplifications.

One crucial identity involving these functions is \( 1 + \tan^2 x = \sec^2 x \). This identity is derived from the Pythagorean identity by dividing each term by \( \cos^2 x \). This relation shows how closely connected these functions are, and they offer alternative ways to express various trigonometric equations.

• The expression \( (\tan x + \sec x)^2 \) can be expanded to \( \tan^2 x + 2\tan x \sec x + \sec^2 x \).
• Identity manipulations frequently involve such expansions and substitutions to transform expressions into favorite forms for verifying equivalencies.

Trigonometric Simplification

Trigonometric simplification involves making a complicated trigonometric expression easier to understand or evaluate. This process is an integral part of verifying identities and solving equations.

A common strategy involves using standard trigonometric identities to transform parts of an expression. For example, simplifying \( \frac{1 + \sin x}{1 - \sin x} \) by using conjugates can lead to easier forms like \( \sec^2 x + 2\tan x \sec x \).

Simplifying trigonometric expressions requires familiarity with multiple identities and transformations:

• Use conjugates to eliminate complex fractions.
• Recognize patterns tied to familiar identities like the Pythagorean identity.
• Translate functions into terms of sine and cosine for increased manipulability.

These skills streamline solving equations and make complex expressions more approachable.