Problem 14 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\frac{\sin ^{2} 2 \alpha}{\sin ^{2} \alpha}=4-4 \sin ^{2} \alpha\)

Step-by-Step Solution

Answer

The identity is verified as both sides equal \(4 \cos^{2} \alpha\).

Step 1: Simplify the Left Side

We begin with simplifying the left side of the equation \( \frac{\sin^{2} 2\alpha}{\sin^{2} \alpha} \). The formula for \( \sin 2\alpha \) is \( 2 \sin \alpha \cos \alpha \). Therefore, \( \sin^{2} 2\alpha = (2 \sin \alpha \cos \alpha)^2 = 4 \sin^{2} \alpha \cos^{2} \alpha \). Thus, the left side becomes \( \frac{4 \sin^{2} \alpha \cos^{2} \alpha}{\sin^{2} \alpha} \).

Step 2: Simplify the Fraction

Cancel \( \sin^{2} \alpha \) from the numerator and the denominator of \( \frac{4 \sin^{2} \alpha \cos^{2} \alpha}{\sin^{2} \alpha} \). This simplifies to \( 4 \cos^{2} \alpha \). Therefore, the left side simplifies to \( 4 \cos^{2} \alpha \).

Step 3: Express Right Side in Terms of Cosine

The identity on the right side \( 4 - 4 \sin^{2} \alpha \) can be rewritten using the Pythagorean identity \( \sin^{2} \alpha + \cos^{2} \alpha = 1 \). So, \( 4(1 - \sin^{2} \alpha) = 4 \cos^{2} \alpha \).

Step 4: Equate the Simplified Expressions

Both sides now simplify to \( 4 \cos^{2} \alpha \). Therefore, the identity is verified as both sides are equal.

Key Concepts

Double Angle Formulas

When dealing with trigonometric expressions, the double angle formulas can be incredibly useful tools. They provide a way to express trigonometric functions of double angles in terms of single-angle functions. For example, the double angle formula for sine is \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \). This formula allows us to simplify expressions involving double angles by breaking them down into simpler parts. In the exercise, we used \( \sin^2 2\alpha \) and expanded it using the double angle formula, which transformed \( \sin^2 2\alpha \) into \((2 \sin \alpha \cos \alpha)^2 = 4 \sin^2 \alpha \cos^2 \alpha \). Understanding and applying these formulas enable us to break down complex trigonometric identities into more manageable forms.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone relationships in trigonometry. It states that for any angle \( \alpha \), \( \sin^2 \alpha + \cos^2 \alpha = 1 \).This identity is not only fundamental but highly useful for simplifying expressions, especially when you need to rewrite one trigonometric function in terms of another. In the given exercise, to simplify the right side of the identity, we used \( \cos^2 \alpha = 1 - \sin^2 \alpha \) by rearranging the Pythagorean identity. This allowed us to rewrite \( 4 - 4 \sin^2 \alpha \) as \( 4 \cos^2 \alpha \).The Pythagorean identity is extremely versatile in various situations involving trigonometric expressions and equations, making it an essential tool for any student studying trigonometry.

Trigonometric Simplification

Trigonometric simplification involves expressing a complex trigonometric expression in a simpler or more convenient form. This process often utilizes identities such as double angle formulas and Pythagorean identities.In the provided exercise, we started by simplifying a fraction involving \( \sin^2 2\alpha \) on the left side. By using appropriate identities, we reduced the expression step by step to reach a more straightforward form. Key strategies include:

• Identifying applicable formulas or identities.
• Cancelling out common terms in fractions.
• Rewriting parts of an expression using known identities.

Through simplifying trigonometric expressions, students gain deeper insights into the relationships between different trigonometric functions and develop skills useful for solving various mathematical problems.