Problem 3 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{\sec ^{2} 2 u-1}{\sec ^{2} 2 u}=\sin ^{2} 2 u$$

Step-by-Step Solution

Answer

The identity is verified: both sides equal \( \sin^2 2u \).

Step 1: Rewrite Identity in Terms of Cosine

Start by rewriting \( \sec^2 2u \) in terms of cosine. Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec^2 2u = \frac{1}{\cos^2 2u} \).The left side becomes:\[ \frac{\sec^2 2u - 1}{\sec^2 2u} = \frac{\frac{1}{\cos^2 2u} - 1}{\frac{1}{\cos^2 2u}} \]

Step 2: Simplify the Fraction

Simplify the expression by finding a common denominator in the numerator:\[ \frac{\frac{1 - \cos^2 2u}{\cos^2 2u}}{\frac{1}{\cos^2 2u}} = \frac{1 - \cos^2 2u}{\cos^2 2u} \times \cos^2 2u = 1 - \cos^2 2u \]

Step 3: Use Pythagorean Identity

Use the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \). Compare this with the expression obtained in Step 2:\[ 1 - \cos^2 2u = \sin^2 2u \]

Step 4: Verify Identity

Since the expression \( 1 - \cos^2 2u \) equals \( \sin^2 2u \), we have verified that the original identity is true. The left side simplifies and matches the right side, confirming the identity as \( \sin^2 2u = \sin^2 2u \).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially in geometry and calculus. These functions relate angles of triangles to ratios of side lengths. The three primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each of these functions has a reciprocal function:

• Secant (\( \sec \)) is the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \).
• Cosecant (\( \csc \)) is the reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \).
• Cotangent (\( \cot \)) is the reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \).

These functions are pivotal in various mathematical identities and equations. They help us simplify expressions and solve trigonometric equations. Understanding how these functions relate to each other provides a foundation for working with identities, like \( \sec^2 \theta \) in this case, which we transformed using \( \sec \theta = \frac{1}{\cos \theta} \). This transformation helps in the simplification processes, leading us to important trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry, expressing a fundamental relationship between sine and cosine. The identity is given by:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This equation is derived from the Pythagorean Theorem and applies to any angle \( \theta \). Two useful forms, obtained by manipulating the identity, are:

• \( 1 - \cos^2 \theta = \sin^2 \theta \)
• \( 1 - \sin^2 \theta = \cos^2 \theta \)

In verifying trigonometric identities, like in this exercise, these forms help manipulate and simplify expressions. For instance, identifying that the expression \( 1 - \cos^2 2u \) matches \( \sin^2 2u \) allowed us to confirm the original identity. This process highlights the power of the Pythagorean Identity in breaking down complex trigonometric equations into simpler components.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a technique used to make complex identities more manageable. The process involves using known identities and algebraic manipulations to reduce expressions into simpler forms. Here are steps often involved:

• Rewriting trigonometric functions in terms of sine and cosine, which are the basic components of many identities.
• Using known identities like the Pythagorean Identity to replace parts of the expression with equivalent terms.
• Finding common denominators to combine fractions effectively, as seen when simplifying the expression \( \frac{1 - \cos^2 2u}{\cos^2 2u} \).

These techniques are applied to arrive at an equivalent, often simpler, expression that matches the desired form. In our example, simplifying revealed the inherent truth of the identity \( \sin^2 2u \). Mastery of such simplifications enhances efficiency and understanding in solving trigonometric problems.