Problem 41 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ 1+\frac{1}{\cos x}=\frac{\tan ^{2} x}{\sec x-1} $$

Step-by-Step Solution

Answer

Identity verified; both sides simplify to \( \frac{1+\cos x}{\cos x} \).

Step 1: LHS Transformation

Start with the left-hand side of the equation: \( 1 + \frac{1}{\cos x} \). Express 1 as \( \cos x / \cos x \) so the expression becomes \( \frac{\cos x + 1}{\cos x} \).

Step 2: Simplify RHS

The right-hand side is \( \frac{\tan^2 x}{\sec x - 1} \). Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \), thus \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \) and \( \sec x - 1 = \frac{1}{\cos x} - 1 = \frac{1 - \cos x}{\cos x} \).

Step 3: Transform RHS Further

Substitute the expressions for \( \tan^2 x \) and \( \sec x - 1 \) from Step 2 into the RHS: \( \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1-\cos x}{\cos x}} = \frac{\sin^2 x}{(1-\cos x)\cos x} \).

Step 4: Use Pythagorean Identity

Use the identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite \( \sin^2 x \) in terms of \( \cos x \): \( \sin^2 x = (1-\cos x)(1+\cos x) \). Substitute this back into the RHS: \( \frac{(1-\cos x)(1+\cos x)}{(1-\cos x)\cos x} \).

Step 5: Simplify Final Expression

Cancel the common factor \((1-\cos x)\) from numerator and denominator: \( \frac{1+\cos x}{\cos x} \).

Step 6: Compare Simplified Expressions

Notice both LHS, \( \frac{1+\cos x}{\cos x} \), and RHS, \( \frac{1+\cos x}{\cos x} \), are identical after simplification, thus verifying the identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a foundational concept in trigonometry. It tells us that for any angle \( x \), the equation \( \sin^2 x + \cos^2 x = 1 \) holds true. This can be incredibly useful when verifying or transforming trigonometric identities. By using this identity, we can express one trigonometric function in terms of another, which often simplifies the original problem.

• For instance, if we need \( \sin^2 x \), we can rewrite it as \( 1 - \cos^2 x \).
• Similarly, \( \cos^2 x \) can be expressed as \( 1 - \sin^2 x \).

Understanding and applying the Pythagorean Identity allows us to simplify expressions and verify identities in various trigonometric problems. In our exercise, it helped transform \( \sin^2 x \) into a product involving \( 1 - \cos x \) and \( 1 + \cos x \), which was a crucial step towards simplification.

Trigonometric Transformation

Trigonometric transformations involve manipulating expressions using known identities and relationships between trigonometric functions, such as tangent, secant, sine, and cosine. Transformations allow us to convert a complex expression or equation into a simpler form or another equivalent form.
In the context of our exercise, several transformations were applied:

• The expression for \( 1 \) was transformed into \( \cos x / \cos x \) to combine with another fraction.
• The identity \( \tan x = \sin x / \cos x \) was used, where \( \tan^2 x \) turned into \( \sin^2 x / \cos^2 x \).
• The transformation continued by turning \( \sec x - 1 \) into \( (1 - \cos x) / \cos x \).

Each transformation saved space and revealed equal or simplified forms of both sides of the given identity. These steps illustrate the power of transformations in reducing complex trigonometric problems to manageable equations that align closely with one another.

Verification of Trigonometric Identities

Verification of trigonometric identities involves showing that two sides of a given equation are equivalent for all values of the involved variables. This involves simplifying both sides of the equation independently until they are shown to be the same.
The process often requires:

• Identifying and applying suitable trigonometric identities, like the Pythagorean Identity.
• Performing algebraic manipulations such as factoring, combining fractions, or canceling common terms.

In the exercise at hand, verification was completed by simplifying both the left-hand side (LHS) and right-hand side (RHS) of the equation to the same form: \( \frac{1 + \cos x}{\cos x} \). This demonstrated that both sides are, indeed, identical for all applicable angles \( x \). Verification reassures that the transformation and identity applications are correct, confirming that our understanding of trigonometric properties is sound.