Problem 114 - Precalculus with Limits

Question

In Exercises 111 - 124, verify the identity. \( \dfrac{\cos 3\beta}{\cos \beta} = 1 - 4 \sin^2 \beta \)

Step-by-Step Solution

Answer

Step 1: Understand that we want to prove the given equation. Step 2: Apply the trigonometric identities to replace \(\cos 3\beta\) with \(4\cos^3\beta - 3\cos\beta\). Step 3: Substitute the Pythagorean identity \( \cos^2\beta = 1 - \sin^2\beta \) into the equation, which simplifies to the right side of the identity. The identity is verified.

Step 1: Understand the goal

The goal is to show that \( \frac{\cos 3\beta}{\cos \beta} \) equals \( 1 - 4 \sin^2 \beta \). We do that by manipulating one side of the equation until it resembles the other side. In this case, it will be easier to start with the left hand side.

Step 2: Apply trigonometric identities

Using the triple angle identity, \(\cos 3\beta = 4\cos^3\beta - 3\cos\beta\), the left side of the equation, \(\frac{\cos 3\beta}{\cos \beta}\), becomes \(4\cos^2\beta - 3\).

Step 3: Substitute the trigonometric identity

Recall the Pythagorean identity, \( \cos^2\beta = 1 - \sin^2\beta \). Substituting this in for \(\cos^2\beta\) in the left side equation \(4\cos^2\beta - 3\) results in \(4(1-\sin^2\beta) - 3 = 4 - 4\sin^2\beta - 3 = 1 - 4\sin^2\beta\) which is equal to the right side of the identity.

Key Concepts

Verifying Trigonometric Identities

Verifying a trigonometric identity involves showing that two different trigonometric expressions are equivalent. This means transform one side of an equation to make it look exactly like the other side, using known trigonometric identities and algebraic manipulations. When verifying an identity, students should familiarize themselves with various strategies such as working with one side of the equation, substituting known identities, and simplifying. It’s important to recognize when to apply these strategies to simplify the expressions effectively.

To verify the identity \( \frac{\cos 3\beta}{\cos \beta} = 1 - 4 \sin^2 \beta \), we follow these steps: start with one side, apply relevant identities such as the triple angle identity for cosine, and substitute in other known identities like the Pythagorean identity to reach the form of the other side of the equation. This logical progression helps students understand the structure and purpose behind each manipulation.

Understanding the Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry. It states that for any angle \(\beta\), the square of the sine plus the square of the cosine of that angle equals one: \( \sin^2\beta + \cos^2\beta = 1 \). This identity is derived from the Pythagorean theorem in a right-angled triangle context. By dividing through by \(\cos^2\beta\), we also get another useful form: \(1 + \tan^2\beta = \sec^2\beta\).

The Pythagorean identity allows us to express sine in terms of cosine and vice versa which is particularly useful in verifying trigonometric identities. In the given exercise, we utilize this identity to substitute \(\cos^2\beta\) with \(1 - \sin^2\beta\). This step is critical as it allows us to express everything in terms of \(\sin\beta\) to match the right side of the provided identity.

Working with Triple Angle Identities

The triple angle identities express the sine or cosine of three times an angle in terms of the sine or cosine of the angle itself. In the case of cosine, the identity is \(\cos 3\beta = 4\cos^3\beta - 3\cos\beta\). These identities can seem daunting due to their complexity, but they are just extensions of double angle and sum of angles identities.

When faced with an equation like \(\frac{\cos 3\beta}{\cos \beta}\), recognizing that the numerator can be transformed by the triple angle identity is crucial. By doing so, we rewrite the left side of the equation in a form that allows us to incorporate the Pythagorean identity. Strategically using these identities in the exercise simplifies the problem to a point where the original and transformed sides match, confirming the identity. Remember, practice with these identities will improve familiarity and make it easier to spot opportunities to apply them in verifying trigonometric identities.