Problem 40 - Algebra 2

Question

Verify that each of the following is an identity. \(\cos (\alpha+\beta)=\frac{1-\tan \alpha \tan \beta}{\sec \alpha \sec \beta}\)

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \cos \alpha \cos \beta - \sin \alpha \sin \beta \).

Step 1: Recall Trigonometric Identities

To verify the given identity, we need to recall the basic trigonometric identities. We know that the cosine of the sum of two angles, \[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta.\] Also, recall that \[\tan \theta = \frac{\sin \theta}{\cos \theta} \text{ and } \sec \theta = \frac{1}{\cos \theta}.\] We will use these identities to transform each side of the equation.

Step 2: Transform the Left Side

Use the identity for cosine of the sum of two angles: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \). This equation will be our left side.

Step 3: Transform the Right Side

The right side of the given identity is \( \frac{1-\tan \alpha \tan \beta}{\sec \alpha \sec \beta} \). Substitute the identities: \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), \( \tan \beta = \frac{\sin \beta}{\cos \beta} \), and \( \sec \theta = \frac{1}{\cos \theta} \). The right side becomes:\[\frac{1-\frac{\sin \alpha}{\cos \alpha} \cdot \frac{\sin \beta}{\cos \beta}}{\frac{1}{\cos \alpha} \cdot \frac{1}{\cos \beta}} = \frac{\cos \alpha \cos \beta (\cos \alpha \cos \beta - \sin \alpha \sin \beta)}{1}.\] Upon simplification, this results in: \[ \cos \alpha \cos \beta - \sin \alpha \sin \beta.\]

Step 4: Compare Both Sides

Now compare the expressions we obtained:- The left side: \( \cos \alpha \cos \beta - \sin \alpha \sin \beta \).- The right side: \( \cos \alpha \cos \beta - \sin \alpha \sin \beta \).Both expressions are equivalent. Therefore, the identity holds true.

Key Concepts

Cosine of Sum Identity

The Cosine of Sum Identity is a fundamental rule in trigonometry. It describes how the cosine of a sum of two angles can be expressed in terms of the sines and cosines of the individual angles. The identity is given by:\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta. \]This identity is incredibly useful in various mathematical problems, as it allows the decomposition of angle sums into simpler, more manageable components. For example, by knowing the cosine and sine of two angles separately, one can easily calculate the cosine of their sum without needing a calculator or lookup table. This makes it easier to solve trigonometric equations and simplifies mathematical proofs.Remember:- Cosine values are found at the projection of the unit circle.- Subtraction in the formula indicates a balance between their product and a change in the angle values.By understanding this identity, you'll have a powerful tool for many trigonometric calculations, making problem-solving more straightforward.

Tangent

The tangent of an angle, denoted as \( \tan \theta \), is a crucial trigonometric function. It represents the ratio of the sine of the angle to the cosine of the angle:\[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \]This function highlights the relationship between the circle's horizontal and vertical components. The tangent is especially useful when dealing with right-angle triangles, as it relates the opposite side to the adjacent side. A few key points about tangent:

• The tangent function is periodic with a period of \(\pi\), meaning that \(\tan(\theta + \pi) = \tan \theta\).
• It has vertical asymptotes wherever the cosine of the angle is zero, reflecting undefined values.

Tangent plays a crucial role in verifying identities, particularly when simplifying expressions or altering equations involving both sine and cosine.

Secant

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. This gives us:\[ \sec \theta = \frac{1}{\cos \theta}. \]Secant helps expand the understanding of trigonometric functions beyond the basic sine, cosine, and tangent. It is fundamental when working with identities that require manipulation of these reciprocal relationships.Key details about secant:

• It is undefined at angles where \( \cos \theta= 0 \), such as \(\theta = \frac{\pi}{2}\), where it has vertical asymptotes.
• Like tangent, the secant function is periodic, with a period of \(2\pi\).

In proofs or manipulations, such as simplifying expressions, understanding the secant function's properties is essential. It helps transition between functions and can simplify complex trigonometric identities.

Proof Verification

Proof verification in trigonometry involves confirming that an equation holds true for all possible values of the variables. For the identity \( \cos (\alpha+\beta) = \frac{1-\tan \alpha \tan \beta}{\sec \alpha \sec \beta}, \) proof verification starts with transforming the known identities.One begins by recalling relevant trigonometric identities (like the cosine of sum and the definitions of tangent and secant). Transform both sides of the equation using these identities:

• The left side, basing on the cosine sum identity.
• The right side, translating tangent and secant to base trigonometric functions.

Upon simplifying, you should find that both sides are equal, thus confirming the identity. This often involves:- Breaking complex expressions down to simpler functions.- Applying basic trigonometric identities efficiently.- Ensuring every step adheres to mathematical principles.By practicing proof verification, students gain a deeper understanding of trigonometric identities and their applications in mathematical reasoning.