Problem 61 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1+\tan ^{2} u}{1-\tan ^{2} u}=\frac{1}{\cos ^{2} u-\sin ^{2} u}$$

Step-by-Step Solution

Answer

The identity is verified, as both sides simplify to \( \frac{1}{\cos^2 u - \sin^2 u} \).

Step 1: Rewrite the Left Side using Trigonometric Identity

Notice on the left side, the expression \( 1 + \tan^2 u \) can be rewritten using the identity \( 1 + \tan^2 u = \sec^2 u \), so the left side becomes \( \frac{\sec^2 u}{1 - \tan^2 u} \).

Step 2: Rewrite the Right Side using Trigonometric Identity

The right side \( \frac{1}{\cos^2 u - \sin^2 u} \) can be rewritten using the difference of squares: \( \cos^2 u - \sin^2 u = (\cos u + \sin u)(\cos u - \sin u) \).

Step 3: Express \( \sec^2 u \) in terms of \( \cos u \) on Left Side

We know \( \sec u = \frac{1}{\cos u} \), thus \( \sec^2 u = \frac{1}{\cos^2 u} \). Substitute this into the left side to get: \( \frac{\frac{1}{\cos^2 u}}{1 - \tan^2 u} \).

Step 4: Simplify \tan^2 u on the Left Side

We know \( \tan^2 u = \frac{\sin^2 u}{\cos^2 u} \). Substitute into the left side to get: \( \frac{\frac{1}{\cos^2 u}}{1 - \frac{\sin^2 u}{\cos^2 u}} = \frac{1}{\cos^2 u - \sin^2 u} \).

Step 5: Confirm Both Sides Match

Now, observe that \( \frac{1}{\cos^2 u - \sin^2 u} \) is the same on both sides of the equation. Thus, the identity is verified.

Key Concepts

Verification of Identities

Verifying trigonometric identities is an essential exercise in mathematics that involves proving two expressions are equivalent by ensuring both sides of an equation are the same. It requires using known identities and algebraic manipulation techniques to make both sides identical. This process is akin to solving a puzzle where each piece (identity) must fit perfectly. In our exercise,

• The identity to verify is \[ \frac{1+\tan ^{2} u}{1-\tan ^{2} u}=\frac{1}{\cos ^{2} u-\sin ^{2} u} \]
• Start by exploring both sides individually and transforming them using identities.
• The task is to ensure both expressions simplify to the same value.

This involves expressing one side in terms of a basic trigonometric function or identity, like sine or cosine, and manipulating it until it resembles the other side. Verification of identities not only enhances algebraic skills but also deepens understanding of how trigonometric functions are interrelated.

Tan and Sec Relationship

Understanding the relationship between tangent and secant is crucial for simplifying trigonometric expressions. In trigonometry, the tangent function \( \tan u \) is related to secant function \( \sec u \) through the Pythagorean identity:

• \( 1 + \tan^2 u = \sec^2 u \)
• This means whenever you encounter \( 1 + \tan^2 u \), you can substitute it with \( \sec^2 u \).

Let's break down these functions:

• \( \tan u \) is the ratio of \( \sin u \) to \( \cos u \).
• \( \sec u \) is the reciprocal of \( \cos u \), or \( \frac{1}{\cos u} \).

The identity \( 1 + \tan^2 u = \sec^2 u \) is derived from the fundamental Pythagorean identity \( \sin^2 u + \cos^2 u = 1 \), highlighting the interconnected nature of trigonometric functions. By re-expressing tangent in terms of secant, complex expressions become manageable, allowing you to progress with fewer steps.

Difference of Squares Identity

The difference of squares identity is an algebraic concept that involves expressions of the form \( a^2 - b^2 \). In trigonometry, applying this identity can simplify or transform expressions involving sine and cosine.
This identity states that \[ a^2 - b^2 = (a+b)(a-b) \]For trigonometric identities, consider terms like \( \cos^2 u - \sin^2 u \). This can be transformed using:

• \( \cos^2 u - \sin^2 u = (\cos u + \sin u)(\cos u - \sin u) \)

In our task, the difference of squares identity helps factor the denominator on the right side of the given trigonometric equation. This manipulation is pivotal as it unveils simpler forms or matches structures needed for verification. By utilizing the difference of squares, you can effortlessly rewrite expressions, making verification tasks clearer and more straightforward.