Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \frac{1}{\tan \alpha+\tan \beta}=\frac{\cos \alpha \cos \beta}{\sin (\alpha+\beta)} $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal.

Step 1: Convert Left Side to Sine and Cosine

We start by transforming the left side of the identity involving tangent functions into sine and cosine. Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \). Thus, the denominator becomes: \[ \frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta} = \frac{\sin \alpha \cos \beta + \sin \beta \cos \alpha}{\cos \alpha \cos \beta} \] So the entire left side is: \[ \frac{1}{\tan \alpha + \tan \beta} = \frac{\cos \alpha \cos \beta}{\sin \alpha \cos \beta + \sin \beta \cos \alpha} \]

Step 2: Simplify to Match the Right Side

Now, notice that the denominator of the fraction on the left side \( \sin \alpha \cos \beta + \sin \beta \cos \alpha \) is exactly the same as \( \sin(\alpha + \beta) \), which is the formula for the sine of a sum. Therefore, we can express the left side as:\[ \frac{\cos \alpha \cos \beta}{\sin(\alpha + \beta)} \] This matches the right side of the given identity, proving the identity is correct.

Key Concepts

Understanding Sine and Cosine

The sine and cosine are fundamental concepts in trigonometry. They are primarily used to describe the ratios of sides in right-angled triangles and the coordinates of points on a unit circle.

- **Sine** (\( \sin \theta \)) of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.- **Cosine** (\( \cos \theta \)) of an angle is the ratio of the length of the adjacent side to the hypotenuse.

These functions are periodic and have a range from -1 to 1. This periodicity and range make them suitable for modeling wave-like phenomena.

When dealing with identities and transformations, such as the one in the exercise, converting expressions involving other trigonometric functions, like tangent, into sine and cosine helps in simplifying and verifying the identities.

Exploring the Sum of Angles

The sum of angles concept in trigonometry allows us to find the sine and cosine of the sum of two angles, \( \alpha \) and \( \beta \).

The formulas are:

• **Sine of the Sum**: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• **Cosine of the Sum**: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

These identities are essential for simplifying complex expressions in trigonometry.

In our exercise, recognizing that \( \sin(\alpha + \beta) \) can be expressed as \( \sin \alpha \cos \beta + \cos \alpha \sin \beta \) was central to proving the identity. Using these formulas, you can often break down complex identities into manageable steps, simplifying equations and allowing for easier verification of trigonometric identities.

The Tangent Function Simplified

The tangent function is another key trigonometric function that relates to sine and cosine. It is defined as the ratio of sine to cosine:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

In the exercise, the identity involves tangent functions that were converted into sine and cosine to facilitate simplification. This transformation is crucial because it allows us to use known identities and simplify complex expressions.

For sum and difference identities, the tangent of the sum \( \tan(\alpha + \beta) \) is given by:

• \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

However, for this problem, substituting tangent with sine and cosine was more straightforward for simplification. The relationship between sine, cosine, and tangent streams from these basic identities and transformations, making it easier to handle trigonometric equations.